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Higher Physics

Unit 1 Our Dynamic Universe

Section 4 Gravitation

North Berwick High School

Department of Physics

Section 4 Gravitation

Note Making Make a dictionary with the meanings of any new words.

The physics of projectiles 1. State what is meant by a projectile.

Horizontal projectiles 1. Describe an experiment to confirm that the horizontal and vertical components of motion are entirely independent of one another. 2. Your teacher will show you how to lay out a horizontal projectile calculation using question 1 on page 24 as an example. Copy this into your notes.

Projectiles at an angle to the horizontal. 1. Show how the values for the horizontal and vertical components of the launch velocities are found. 2. Copy the summary table on page 8 for any projectile. 3. Your teacher will show you how to layout an angled projectile calculation using question 6 on page 26 as an example. Copy this into your notes.

Projectiles and Newton’s thought experiment 1. Describe Newton's thought experiment. 2. Consider the questions on page 11. Are you able to answer any? (There is no need to make a note.)

The law of universal gravitation 1. State Newton's theory of universal gravitation. 2. Copy the formula along with the units. 3. Copy the examples on pages 17 - 19.

The force of gravity on a planetary scale 1. Write a note explaining gravity assist. You might find the tennis ball analogy useful.

The force of gravity on a universal scale 1. Describe how a star is born within a nebula. 2. Describe how planets may now be formed around the star. 3. Consider the same questions again on page 23. Are you able to answer any now? (Still no need to make a note.) Think about the next set of questions on page 23.

Section 4 Gravitation

Contents

Content Statements ......................................................................... 1

The physics of projectiles ................................................................. 2

Projectiles and Newton’s thought experiment .................................. 9

Understanding gravity .................................................................... 10

Mapping the gravitational field on Earth ........................................ 12

The Earth’s natural satellite: the Moon .......................................... 14

Evidence in support of the geocentric model .................................. 14

Evidence in support of a new model – heliocentrism ...................... 15

The Moon – what keeps it in place? ............................................... 15

The law of universal gravitation ..................................................... 16

The force of gravity in everyday life ............................................... 17

The force of gravity on a subatomic scale ....................................... 19

The force of gravity on a planetary scale ........................................ 20

The force of gravity on a universal scale ......................................... 21

A huge cloud of dust ...................................................................... 22

A star is born ................................................................................. 22

The formation of the planets .......................................................... 22

Problems ....................................................................................... 24

Solutions ........................................................................................ 29

1

Content Statements

Content Notes Context

a) b)

Projectiles and Satellites. Gravity and mass.

Resolving the motion of a projectile with an initial velocity into horizontal and vertical components and their use in calculations. Comparison of projectiles with objects in free fall. Newton’s thought experiment and an explanation of why satellites remain in orbit. Gravitational Field Strength of planets, natural satellites and stellar objects. Calculating the force exerted on objects placed in a gravity field. Newton’s Universal Law of Gravitation.

Using software to analyse videos of projectiles Low orbit and geostationary satellites. Satellite communication and surveying. Environmental monitoring of the conditions of the atmosphere. Methods for measuring the gravitational field strength on Earth. Using the slingshot effect to travel in space. Lunar and planetary orbits. Formation of the solar system by the aggregation of matter. Stellar formation and collapse. The status of our knowledge of gravity as a force may be explored. The other fundamental forces have been linked but there is as yet no unifying theory to link them to gravity.

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Section 4 Gravitation

The physics of projectiles

© Gustoimages/Science Photo Library

A projectile is an object on which the only force acting is gravity. There are a variety of examples of projectiles: an object dropped from rest is a projectile (provided that the influence of air resistance is negligible), an object thrown vertically upwards is a projectile (provided that the influence of air resistance is negligible) and an object thrown upwards at an angle is also a projectile (the same assumption). A projectile is any object, which, once projected, continues its motion by its own inertia and is influenced only by the downward force of gravity. By definition, a projectile has only one force acting on it: the force of gravity. If there were any other force acting on an object, then that object would not be a

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projectile. Projectiles can be launched horizontally, vertically or at an angle between the two. All projectiles have a horizontal and vertical velocity. They also have a horizontal and vertical displacement. The equations of motion can be used to describe the motion of projectiles. You have already investigated one type of projectile: a vertical projectile. This is the case of the tennis ball allowed to fall freely under the force of gravity, or the tennis ball allowed to fall freely under the force of gravity and allowed to bounce, or the tennis ball thrown up in the air and allowed to return to its starting position. To summarise, for a vertical projectile:

Direction of motion

Forces Velocity Acceleration

Horizontal

Air resistance negligible so no forces in the horizontal

Constant (in this case 0 m s–1)

None

Vertical

Air resistance negligible so only force of gravity acting in the vertical

Changing with time

Constant or uniform acceleration (– 9.8 m s-2)

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The second projectile situation to consider is the horizontal projectile. Picture a motorcyclist on the top of a tall building about to perform a death-defying stunt of incredible skill. Do not try this at home! Predict her path as she drives across the roof at speed and off the edge.. Explain your prediction. You should investigate horizontal projectiles to confirm that the horizontal and vertical components of motion are entirely independent of one another. This will also give you an opportunity to verify the equations of motion, i.e. to confirm that the equations we use correctly describe the motion that happens in real life. Some of the experiments you might experience include:

filming motion of a horizontal projectile and confirming the horizontal and vertical motion using software such as tracker.jar

using a traditional piece of equipment that allows you to compare the time of flight for an object dropped vertically and an object fired horizontally, both being released simultaneously and from the same height

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using a thought experiment and a spreadsheet programme to calculate and graph horizontal displacement against vertical displacement for an object dropped vertically and an object fired horizontally, both being released simultaneously and from the same height

using a ballistics cart, e.g. PASCO

using a projectile launcher and timing system to confirm that the time of flight recorded matches that which can be determined theoretically, estimating and accounting for experimental uncertainties.

To summarise, for a horizontal projectile:

Direction of motion

Forces Velocity Acceleration

Horizontal

Air resistance negligible so no forces in the horizontal

Constant None

Vertical

Air resistance negligible so only force of gravity acting in the vertical

Changing with time

Constant or uniform acceleration (– 9.8 m s-2)

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The third and final projectile situation to consider is the projectile at an angle to the horizontal.

© Gustoimages/Science Photo Library

Remember that any vector can be resolved into its horizontal and vertical components: cos θ = adjacent hyp adjacent = hyp cos θ horizontal component = launch velocity cos θ

launch velocity (ms-1)

θ

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sin θ = opposite hyp opposite = hyp sin θ vertical component = launch velocity sin θ The distance travelled horizontally (the range) is determined by the cosine component of the launch velocity. The time of flight is determined by the sine component of the launch velocity. Determine experimentally and/or theoretically the angle, which gives the greatest range for a fixed, launch velocity. Your teacher will show you how to lay out the answer to question 1 on page 24.

θ

launch velocity (ms-1)

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To summarise, for a projectile at an angle to the horizontal:

Direction of motion

Forces Velocity Acceleration

Horizontal

Air resistance negligible so no forces in the horizontal

Constant None

Vertical

Air resistance negligible so only force of gravity acting in the vertical

Changing with time

Constant or uniform acceleration (– 9.8 m s-2)

To summarise, for any projectile:

Direction of motion

Forces Velocity Acceleration

Horizontal

Air resistance negligible so no forces in the horizontal

Constant None

Vertical

Air resistance negligible so only force of gravity acting in the vertical

Changing with time

Constant or uniform acceleration (– 9.8 m s-2)

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Projectiles and Newton’s thought experiment Think back to our death-defying motorcyclist. Consider what would happen if: the building were taller the horizontal launch velocity were greater the Earth curved away more steeply.

This is what Newton (1643–1727) thought about. He considered that there is a horizontal launch velocity that would result in the projectile continually falling to Earth as a result of the force of gravity at the same rate at which the Earth’s surface curves away beneath it. The projectile would continually fall but would not hit the surface. It would therefore orbit the Earth. His thought experiment explained satellite motion around 300 years before it became a reality. On October 4 1957 Newton’s thought experiment became reality with the successful launch of the world’s first artificial satellite, Sputnik. The significance of this launch in terms of its impact on life cannot be underestimated. Low -orbit and geostationary satellites are used for a wide range of applications, including environmental monitoring, military applications and defence, communications, and scientific investigations.

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Understanding gravity Sir Isaac Newton (1642–1727) did not discover gravity. People had long been aware that objects when released would fall.

www.Cartoonstock.com

Galileo (1564–1642) had carried out experiments on falling objects and put forward a theory on the motion of objects in freefall in his unfinished work De Motu (On Motion). Obviously, then, Galileo was performing experiments at the very beginning of his investigations into motion, and he took his experimental results seriously. Over two decades he changed his ideas and refined his experiments, and in the end he arrived at the law of falling bodies which states that in a vacuum all bodies, regardless of their weight, shape, or specific gravity, are uniformly accelerated in exactly the same way, and that the distance fallen is proportional to the square of the elapsed time. The Galileo Project: http://galileo.rice.edu/sci/theories/on_motion.html

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Whilst not the first to investigate gravity,Newton’s significant contribution, ranked ‘among humanity’s greatest achievements in abstract thought’ (http://www.newton.ac.uk/newtlife.html), was to theorise that the force acting locally on an apple could be applied to the universe. Newton developed the universal theory of gravitation. This was published in his best-known work, the Principia.

http://www.newtonprject.sussex.ac.uk

Questions to consider before you progress through this section:

What is gravity?

What is the force of gravity?

What are the effects of gravity?

What do we know about gravity?

How can we make use of gravity?

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Mapping the gravitational field on Earth On 17 March 2009, the European Space Agency launched the Gravity Field and Steady State Ocean Circulation Explorer (GOCE), the lowest orbit research satellite in operation (at 254.9 km).

© ESA

(see http://www.esa.int/esaLP/ESAYEK1VMOC_LPgoce_0.html).

The mission of this satellite is to map the Earth’s gravitational field in greater detail than has previously been possible. This data wi ll be used to:

inform predictions of climate

understand and monitor the effects of climate change, making accurate measurements of ocean circulation and sea level

develop an improved understand of the internal geology of the Earth, including hazards such as volcanoes

develop a system that allows comparison of heights all over the world to be made for the first time; such a system would be very useful when undertaking large-scale engineering projects such as bridge building and tunnel construction.

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© ESA

In order to achieve its very challenging mission objectives, the satellite was designed to orbit at a very low altitude, where the gravitational variations are stronger closer to Earth. Since mid-September 2009, GOCE has been in its gravity-mapping orbit at a mere 254.9 km mean altitude – the lowest orbit sustained over a long period by any Earth observation satellite. The residual air at this low altitude causes the orbit of a standard satellite to decay very rapidly. GOCE, however, continuously nullifies the drag in real time by firing an ion thruster using xenon gas. It ensures the gravity sensors are flying as though they are in pure freefall, so they pick up only gravity readings and not the disturbing effects from other forces. To obtain clean gravity readings, there can be no disturbances from moving parts, so the entire satellite is a single extremely sensitive measuring device.

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‘The gravity measuring system is functioning extremely well. The system is actively compensating for the effects of atmospheric drag and delivering a continuous set of clean gravity readings,’ Dr Floberghagen a Physicist working on it said. ‘This in itself is an excellent technical achievement. GOCE has proven to be a nearly perfect satellite for measuring gravity from space.’ (Extract from http://www.esa.int/SPECIALS/GOCE/SEMY0FOZVAG_0.html) The ‘standard’ acceleration due to gravity at the Earth’s surface is 9.8 m s –2. The GOCE mission has established that the figure in fact varies from 9.788 m s –2 at the equator to 9.838 m s–2 at the poles.

The Earth’s natural satellite: the Moon Greek philosophers understood that the Moon is a sphere in orbit around the Earth. They also realised that the Moon is not a light source, but reflects sunlight. Around 1850 years ago, Ptolemy (90–168) hypothesised that the Moon and Sun both orbit the Earth. This theory was the most commonly held belief in ancient Greece, Europe and in many other parts of the world. There is evidence that others, including Muslim scholars, developed an alternative theory, one in which the Earth was not orbited by the Sun, 250 years before Ptolemy proposed his theory.

Evidence in support of the geocentric model Geocentrism is the belief that the Earth is the centre of the universe. What evidence supported this theory? What evidence do we now have to support an alternative theory? Certainly, the Sun, the Moon, the stars and other planets appear to revolve around us. In fact, we still commonly talk about the Sun in these terms: ‘rising’, ‘setting’, and ‘going down’. These suggest a belief that the Sun is orbiting the Earth. The Ancient Greeks also believed that the Earth is stationary. How do you know it isn’t?

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Evidence in support of a new model – heliocentrism With the work of Copernicus (1473–1543) the prevailing view of the universe began to change. Kepler (1571–1630) developed three laws that predicted that the orbits of the planets are elliptical, with the Sun at a focus of the ellipse. What evidence was there to support this model at the time when Copernicus and Kepler were developing these theories and laws? What evidence i s there to support this model now?

The Moon – what keeps it in place? The Moon remains in orbit around the Earth as a result of the force of gravity. It is the weakest of the four fundamental forces (the others being the strong force, the weak force and the electromagnetic force) yet it keeps the universe in shape! Newton developed the theory of universal gravitation. This was a very important piece of work – not least because he proposed it to be universal, i.e. he proposed that all parts of the universe obey the same laws of nature. His theory hypothesised (and proved mathematically) that the Moon orbits the Earth as a result of the same force that causes an apple to fall from a tree. The moon falls around the Earth. What evidence did Newton have available to him to support this theory? What evidence do we now have? As we have gathered more and more scientific evidence about the universe, we have come to take for granted the universality of the laws of nature.

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The law of universal gravitation Newton’s law of universal gravitation proposed that each body with mass will exert a force on each other body with mass. The theory states that the force of gravitational attraction is dependent on the masses of both objects and is inversely proportional to the square of the distance that separates them. But Newton remained uncertain. He could not account for action at a distance without some medium, i.e. he was concerned about the distance over which he proposed gravitational force would act in space and the fact that space is a vacuum.

1 2

2

Gm mF

r

F is force in newtons (N) m1 and m2 are the two masses measured in kilograms (kg) r is the distance between them (m) G is the gravitational constant N m2 kg–2

The value of the gravitational constant was determined by Cavendish (1731–1810) in the late 1700s. It was another hundred years before Boys (1855–1944) improved on its accuracy. G = 6.67428 × 10–11 N m2 kg–2

G remains one of the most difficult constants to measure with accuracy. In 2007 a further value was published which suggested an improvement on the accuracy: G = 6.67 × 10–11 N m2 kg–2 is the value we will use for calculations in Higher

Physics.

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The force of gravity in everyday life Newton’s law of universal gravitation allows us to calculate the force of gravity between point masses or spherical objects. Consider two objects, a folder of mass 0.3 kg and a pen of mass 0.05 kg, placed on a desk, 0.25 m apart. Calculate the magnitude of the gravitational force between the two masses (noting that the force is always attractive). Assume that the objects can be approximated to spherical objects. F = ? m1 = 0.30 kg m2 = 0.05 kg r = 0.25 m

G = 6.67 × 10–11 N m2 kg–2

1 2

2

Gm mF

r

–11

2

11

6.67 10 0.30 0.05

0.25

1.60 10

F

F N The gravitational force always acts in a straight line between the two objects being considered. Note that the gravitational force is always attractive, unlike electrostatic and magnetic forces.

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Consider the gravitational force due to the Earth acting on the pen. Assume that the Earth and pen can be approximated to spherical objects. F = ? m1 = 5.97 × 1024 kg m2 = 0.05 kg r = 6.38 × 106 m

G = 6.67 × 10–11 N m2 kg–2

1 2

2

Gm mF

r

–11 24

6 2

6.67 10 5.97 10 0.05

(6.38 10 )

0.489N

F

F How else could this calculation have been carried out? W = mg W = 0.05 × 9.8 W = 0.49 N

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The force of gravity on a subatomic scale Consider the attractive gravitational force between two protons in a nucleus.

1 2

2

Gm mF

r

F is force in newtons (N) = ? m1 = 1.67 × 10–27 kg (mass of proton) m2 = 1.67 × 10–27 kg (mass of proton) r = 0.84 × 10–15 m (radius of proton; radius of neutron assumed equal) G = 6.67 × 10–11 N m2 kg–2

F = 6.67 × 10–11 x 1.67 × 10–27 x 1.67 × 10–27 (0.84 × 10–15)2 F = 2.64 × 10–34 N The force of gravity is clearly insignificant except where we are dealing with very large mass. At the subatomic scale, and at very short range, the strong force is the most significant. The electromagnetic (repulsive) force between two protons is 3.27 x 10-16 N. However, although the weakest of the four forces, the gravitational force acts over enormous distances and on a universal scale!

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The force of gravity on a planetary scale The theory of universal gravitation can be used beyond satellite motion. It is commonly used in space travel in a technique called gravity assist. The gravity assist technique is often called, incorrectly, the ‘slingshot’ effect. The physics of the slingshot effect are considerably more complex. Gravity assist was made famous in the movie ‘Apollo 13’. The craft had been irreparably damaged by an explosion on board. The planned Moon landing was abandoned and the priority became the safe return of the astronauts to Earth. Gravity assist was the technique by which relatively small amounts of the limited available fuel could be used to manoeuvre the craft onto a trajectory that would allow the Moon’s gravitational field to turn the ship. Detailed information on the mission and the problems encountered is available from NASA (see http://science.ksc.nasa.gov/history/apollo/apollo-13/apollo-13.html). Gravity assist is routinely used to boost space flight on unmanned missions to distant planets such as Jupiter and Venus. To understand the principles of this, let us first consider a simple analogy. Consider a tennis ball travelling towards a tennis player holding her racquet at the ready. In this analogy, the ball is the spacecraft and the racquet is a massive planet. The racquet hits the ball. We will consider the outcome to be that the player successfully hits the ball, causing it to change direction and increase speed. Some questions (and answers): What happens to the ball? (it changes direction and increases speed) What type of interaction is this? (a collision) What quantity is conserved in all collisions? (momentum) What other quantity is considered in collisions? (energy) In this case, what can we say about the momentum of the tennis ball? (it increases in magnitude) And the momentum of the racquet? (the law of conservation of momentum tells us it must decrease in magnitude) What about the energy of the tennis ball? (it increases) What is the observable effect of the change in momentum of the tennis ball? (an observable change in speed)

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Is there an observable effect of the change in momentum of the racquet? (no – because the racquet has a much larger mass than the ball) Is there an effect of the change in momentum of the racquet? (yes – it slows down) In this case we have considered a mechanical interaction. The gravity -assist method is a gravitational exchange between the planet and the spacecraft. The spacecraft, which has very small mass, is able to gain momentum and energy from the planet, which has enormous mass. The mass difference is such that whilst the effect on the spacecraft (i.e. an increase in momentum and therefore speed) is observable and useable, the effect on the planet (i.e. a decrease in momentum and therefore speed) is not observable and in fact is negligible. Gravity assist makes use of the universal law of gravitation to enable space flight with minimal fuel requirements. This reduces the weight of the craft and allows us to reach greater distance than would otherwise be possible.

The force of gravity on a universal scale We have explored the effects of the force of gravity on a small scale, its importance in satellite motion and its use in space flight. We have discussed some of the historical story associated with our understanding of gravitational force, but we have yet to discuss a very significant impact of gravitational force. Much of the current scientific thinking is that gravitational force is responsible for the formation of the solar system by aggregation (or accretion) of matter. Gravitational force also plays a crucial role in the birth and death of stars. The solar system formed around 4.5 billion years ago from a huge swirlin g cloud of dust. We know this because advances in technology, such as the Hubble telescope, have allowed us to look deep into space to observe the birth of stars similar to our sun.

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A huge cloud of dust Throughout the Milky Way, and other galaxies like it, are gigantic swirling clouds of dust and gas known as nebulae. It is within nebulae that stars are born. Our star, the Sun, was created in one such nebula. Something, perhaps the shock wave from an exploding supernova (dying star), triggered dust particles to be drawn together to form a dense spherical cloud. The accumulation of dust set off a chain reaction. As the core of the cloud attracted more dust, its gravitational pull increased. More and more dust was sucked in and the cloud collapsed in on itself. As this happened, the rotation of the cloud increased in speed, as happens when spinning ice-skaters pull in their arms. The rotational forces at the equator of the cloud prevented dust along this plane being drawn in, causing the cloud to flatten into a disc spinning around a dense core.

A star is born As more and more mass is accumulated at the centre of the disc, the temperature increased dramatically. Eventually there was enough energy to set off nuclear reactions. Hydrogen atoms fused to form helium, releasing enormous amounts of energy in vigorous bursts. This marked the birth of the Sun, although it would take between 1 and 10 million more years for it to settle into the main sequence star recognisable today.

The formation of the planets The planets, and other extraterrestrial objects such as asteroids, formed in the flat plane of the spinning disc of dust. Electrostatic forces or sticky carbon coatings made dust particles stick together to form clusters, which in turn stuck together to form rocks. Mutual gravity caused these rocks to come together, eventually to form planets. This ‘coming together’ of material is a process known as accretion (or aggregation).

A pillar of dust and gas in the Orion Nebula. © NASA

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Let us now return to the questions to consider:

What is gravity?

What is the force of gravity?

What are the effects of gravity?

What do we know about gravity?

How can we make use of gravity? Are we now in a position to answer these questions conclusively? The answer of course is no. We have explored the force of gravity and some uses of it. However, 350 years after Newton identified gravitational force as a ‘universal’ force; it remains the least understood of the four fundamental forces. We have evidence of its role in the formation of the solar system, but do not conclusively know that this theory is the correct one. We know that it is associated with mass but we do not understand the mechanism for this. In fact there are many questions still unanswered and more research and experimentation to be done.

What effect does Gravity have on space-time?

Is the graviton the exchange particle of gravitational fields?

Is G, the gravitational constant, really constant and is it really universal?

What is the nature of mass?

What are Black holes (and white holes?).

What is the universe made of?

Why does the gravitational mass of galaxy exceed the mass of the known matter? Is there something else there? Or do we not really understand gravity at all?

When were first stars formed? What were they like?

Do gravitational waves exist?

Why do the four fundamental forces have the strengths they have?

Is there a single unifying theory which links gravitational force to the other much better understood forces?

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Gravitation Problems

Projectiles (Revision from Section 1)

1. A plane is travelling with a horizontal velocity of 350 m s 1 at a height of 300 m. A box is dropped from the plane.

The effects of friction can be ignored. (a) Calculate the time taken for the box to reach the ground. (b) Calculate the horizontal distance between the point

where the box is dropped and the point where it hits the ground.

(c) What is the position of the plane relative to the box when the box hits the ground?

2. A projectile is fired horizontally with a speed of 12·0 m s 1 from the edge of a cliff. The projectile hits the sea at a point 60·0 m from the base of the cliff.

(a) Calculate the time of flight of the projectile. (b) What is the height of the starting point of the projectile

above sea level? State any assumptions you have made.

3. A ball is thrown horizontally with a speed of 15 m s 1 from the top of a vertical cliff. It reaches the horizontal ground at a distance of 45 m from the foot of the cliff.

(a) (i) Draw a graph of vertical speed against time for the

ball for the time from when it is thrown until it hits the ground.

(ii) Draw a graph of horizontal speed against time for the ball.

(b) Calculate the velocity of the ball 2 s after it is thrown. (Magnitude and direction are required.) 4. A football is kicked up at an angle of 70° above the horizontal at

15 m s 1. Calculate: (a) the horizontal component of the velocity (b) the vertical component of the velocity.

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5. A projectile is fired across level ground and takes 6 s to travel from A to B. The highest point reached is C. Air resistance is negligible.

Velocity-time graphs for the flight are shown below. VH is the horizontal

velocity and VV is the vertical velocity.

(a) Describe: (i) the horizontal motion of the projectile (ii) the vertical motion of the projectile. (b) Use a vector diagram to find the speed and angle at

which the projectile was fired from point A. (c) Find the speed at position C. Explain why this is the

smallest speed of the projectile. (d) Calculate the height above the ground of point C. (e) Find the horizontal range of the projectile.

VV /m s-1

40

6 3

3 6 0 0

30

-30

time /s time /s

VH /m s-1

C

A B

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6. A ball of mass 5·0 kg is thrown with a velocity of 40 m s 1 at an angle of 30°

to the horizontal.

Calculate: (a) the vertical component of the initial velocity of the ball (b) the maximum vertical height reached by the ball (c) the time of flight for the whole trajectory (d) the horizontal range of the ball.

7. A launcher is used to fire a ball with a velocity of 100 m s 1 at an angle of 60° to the ground. The ball strikes a target on a hill as shown.

(a) Calculate the time taken for the ball to reach the target. (b) What is the height of the target above the launcher?

40 ms-1

30°

60°

100 ms-1

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8. A stunt driver attempts to jump across a canal of width 10 m. The vertical drop to the other side is 2 m as shown.

(a) Calculate the minimum horizontal speed required so that

the car reaches the other side. (b) Explain why your answer to (a) is the minimum horizontal

speed required. (c) State any assumptions you have made. 9. A ball is thrown horizontally from a cliff. The effect of friction can be

ignored. (a) Is there any time when the velocity of the ball is parallel

to its acceleration? Justify your answer. (b) Is there any time when the velocity of the ball is

perpendicular to its acceleration? Justify your answer. 10. A small ball of mass 0·3 kg is projected at an angle of 60° to the horizontal.

The initial speed of the ball is 20 m s 1.

Show that the maximum gain in potential energy of the ball is 45 J.

11. A ball is thrown horizontally with a speed of 20 m s 1 from a cliff. The effects of air resistance can be ignored. How long after being thrown will the velocity of the ball be at an angle of 45° to the horizontal?

60o

20 m s-

1 20 ms-1

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Gravity and mass In the following questions, when required, use the following data:

Gravitational constant = 6·67 × 10 11 N m2 kg 2

1. State the inverse square law of gravitation. 2. Show that the force of attraction between two large ships, each of mass

5·00 × 107 kg and separated by a distance of 20 m, is 417 N. 3. Calculate the gravitational force between two cars parked 0·50 m apart.

The mass of each car is 1000 kg. 4. In a hydrogen atom an electron orbits a proton with a radius of 5·30 ×

10 11 m. The mass of an electron is 9·11 × 10 31 kg and the mass of a

proton is 1·67 × 10 27 kg. Calculate the gravitational force of attraction between the proton and the electron in a hydrogen atom.

5. The distance between the Earth and the Sun is 1·50 × 1011 m. The mass of

the Earth is 5·98 × 1024 kg and the mass of the Sun is 1·99 × 1030 kg. Calculate the gravitational force between the Earth and the Sun.

6. Two protons exert a gravitational force of 1·16 × 10 35 N on each other.

The mass of a proton is 1·67 × 10 27 kg. Calculate the distance separating the protons.

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5. 3·53 × 1022 N

6. 4·00 × 10 15 m

Solutions Projectiles 1. (a) 7·8 s (b) 2730 m 2. (a) 5·0 s (b) 123 m

3. (b) 24·7 m s 1 at an angle of 33° below the horizontal

4. (a) vhoriz = 5·1 ms 1, b) vvert = 14·1 m s 1

5. (b) 50 m s 1 at 36.9° above the horizontal

(c) 40 m s 1 (d) 46 m (e) 240 m

6. (a) 20 m s 1 (b) 20 m (c) 4 s (d) 140 m 7. (a) 8 s (b) 379 m

8. (a) 15·6 m s 1 11. 2 s

Gravity and mass

1. F =

1 2

2

m mG

r

3. 2·67 × 10 4 N

4. 3·61 × 10 47 N