Projectile Motion

11.7 Exercises

1 A ball is thrown horizontally from a cliff with a speed of 10ms-I, at the same time as an identical ball is dropped from the cliff. Neglecting the effect of air resistance and taking gravitational acceleration to be g = +9-8m~-~, sketch graphs (on the axes below) to show

(1) the horizontal speed v, of the projected ball versus time, for the duration of its flighi;

(2) the vertical velocity v,, of the projected and dropped balls vs. time, for the duration of its flight.

On the same axes, sketch graphs to show the horizontal speed and the vertical velocity versus time if air resistance has a significant effect on the flight of the ball. Label each graph clearly to distinguish it from the previous graph.

2 On each of the two projectile trajectories depicted below, draw vectors to show the directions and relative magnitudes of the velocity and the acceleration of the projectile at each of the points A, B & C.

3 (1) If a ball is dropped near the surface_ of the Earth, how long will it take to fall 0.7m?

(2) A tennis player hits a ball horizontally at 35ms-l, when it is 1.6m above the ground. If the ball is hit 1 1-Om from the net will the ball clear the net which is 0-9m high?

10 Physics Essentials Stage 2 Physics

The multiple image photograph on the right shows the motion of two balls that are released at the same time fiom the point 0. Ball A falls freely while ball B is projected horizontally.

(1) Using measurements fiom the diagram, explain how it shows that the horizontal velocity of B remains constant.

(2) Explain how we can use the photograph to deduce that the velocity of ball A increases during its fall.

..---".-"-"-.- --- "-- ---

(3) Explain why ball B has no acceleration in the horizontal direction.

(4) Ball A is falling freely and has a vertical acceleration g = 9 4 r n ~ - ~ . Using the photograph determine the vertical acceleration of ball B. Explain how you got your answer from the photo.

( 5 ) Which ball will hit the ground with the greater speed? Explain your answer.

Projectile Motion

5 (1) An object is dropped from a height of 120m above the Earth.

(a) Find its velocity two seconds after it was dropped.

(b) Find the time taken for it to hit the ground.

(2) A parcel is to be dropped from an aeroplane to a boat at sea. The aeroplane is flying with a speed of looms-' at a fixed altitude of 120m above sea level.

(a) What is the vertical velocity of the parcel after two seconds?

(b) Determine the velocity of the parcel after two seconds.

(c) Explain what happens to the two components of the velocity of the parcel as it falls to the water?

(d) How long, from the moment that it is released, does it take for the parcel to hit the water.

(e) How far, before the aeroplane passes over the boat, from the boat must the parcel be released so that it lands in the water near the boat?

12 Physics Essentials Stage 2 Physics

6 At a point on the upward path of a projectile the velocity of the projectile is 18ms-I at 40" above the horizontal, as shown in the diagram.

(1) Find the horizontal and vertical components of the velocity at this point.

(2) Describe how (and explain why) these components of velocity will change over the rest of the flight.

- --

(3) What is the velocity of the projectile at point X?

7 (1) An object is dropped from a height of 2m above the ground. Find the time that it will take for this body to hit the ground.

-"-" " " -.--- "" "- " ................... " - ........ " ......

(2) A gun, aimed horizontally, fires a bullet with a speed of 900ms". The gun is 2m above ground level.

(a) What is the time of flight of this bullet? Explain your answer.

(b) Find the range of the bullet.

Projectile Motion

(c) Find the velocity with which the bullet hits the ground.

8 A mortar shell is fired from ground level (at point A on the diagram) with a velocity vo = looms-' at an angle of 80' above the horizontal.

(1) Calculate the horizontal and vertical components of the velocity of the shell at the instant it is fired.

(2) Calculate the vertical component of the velocity (i) one second and (ii) thirteen seconds after firing.

(ii)

(3) Calculate the resultant velocity of the shell after 13 seconds.

(4) What is the velocity of the shell at point R?

( 5 ) What is the acceleration of the shell at point B?

Physics Essentials Stage 2 Phvsics

9 (1) An object is dropped from a height of 40m above the surface of the Earth. Calculate long it takes before the object hits the ground?

(2) A stone of mass 200g is thrown with a velocity of vo = 30ms-I horizontally from the observation deck of a lighthouse. At the moment of release the stone is 40m above sea level.

(a) How long does it take before the stone hits the water?

(b) What is the vertical velocity of the stone on impact with the water?

(c) What is the velocity of the stone on impact with the water?

.. .... .----- -- .... " - " " - " ........................

(d) How far does the stone travel horizontally from the point of projection before it hits the water?

(e) Find the total energy (gravitational potential energy and kinetic energy) of the stone at the moment of projection and show that energy is conserved during its flight.

Prdecfile Motion

10 A golfer hits a ball from an elevated tee, at a height of 20m above the green. The ball is hit

..................................................... with a velocity of 50ms-' at an angle of 20' to the horizontal, and its time of flight before it hits the green is 4-41s.

Find the horizontal and components

of the initial velocity.

................... " .... " .................. " ...................... " ....... " ...... " " ........ - "" " "

(2) Find the time that the ball is in the air before it reaches point B, which is at the same height as the tee.

---

(3) Find the distance that the ball travels horizontally before it hits the green.

-

(4) Find the velocity of the ball when it hits the green.

(5) Explain why the kinetic energy of the ball at impact with the green is greater than it is at the instant when it is hit by the golf club.

16 Physics Essentials Stage 2 Physics

11 Water leaves a hose at a speed of va = 2.0ms-', at an angle of 45' above the horizontal. The nozzle is 1 -2m above ground level. The time interval from when the water leaves the nozzle to when A ~r hits the ground is 2.96s.

4 0 . . .. . .. , .. . .. . ... .. . 1 .. . .. .. . . . .. (1) Determine the horizontal distance from the nozzle to the

point where the water hits the ground.

(2) Determine the velocity and the kinetic energy of 1 gram of water at the maximum height.

(3) At which point on the flight path will the speed of the water be the same as the initial speed?

---

(4) The kinetic energy of the water when it hits the ground is greater than when it leaves the nozzle. Explain why this is so and calculate the increase in kinetic energy of one kilogram of water.

(5) What will be the effect on the range if the angle between the nozzle and the horizontal is slightly increased? Explain your answer.

(6) What will be the effect on the range if the angle between the nozzle and the horizontal is slightly decreased? Explain your answer.

Prdectile Motion

12 A cannonball is fixed at an angle of 45" to the horizontal, thus achieving its maximum range of 5297m on horizontal ground. The cannon ball has a flight time of 32-88s. Assume that the cannonball is projected from ground level.

1 5297m (1) Find the horizontal component 7

of the velocity of the cannon ball during its flight.

............... " .......... " - ........... ...... ............................ " - ............ " " "." ............. """ . " ....--.... " ................................................ .......... " ...........................................

(2) Find the initial velocity of the cannonball.

-

(3) What is the initial vertical velocity of the cannonball?

(4) What is the speed of the cannonball at the top of its flight path?

........................................... ..................................... ......... -.- ......................... --..- ......................

(5) Find the time taken for the cannonball to reach its maximum height.

( 6 ) In another identical firing of this cannon, the cannonball encounters a horizontal headwind (i.e. there is no effect on its vertical velocity, only the horizontal speed is reduced). Explain what effect this wind will have on (i) the time of flight and (ii) the range of the cannonball.

1 8 Physics Essentials Stage 2 Physics

13 In an investigation into projectile motion, students projected a golf ball from ground level with the same initial speed but at different angles to the horizontal. At their 'rst attempt the ball was projected at 45" to the horizontal and the range was noted. In subsequent attempts, the angle of projection was progressively increased. Explain what effect (if any) increasing the angle of projection has on

(1) the ball's time of flight.

- -- -

(2) the horizontal component of the ball's initial velocity.

. .. -- ---------------.- ---.-

(3) the range of the ball.

14 An athlete, competing in a shot put event, throws a shot of mass 6.0kg with an initial speed of 13111s-' at an angle of 40" to the horizontal. The shot is released at a height of 2.0m above ground level.

(1) Calculate the time taken for the shot to reach its maximum height of 5.56m above the ground.

--.." ----------------.- .............

(2) Calculate the tirne taken for a body dropped from 5.56m above the Earth to fall to the ground.

(3 ) Using the results of (1) and (2) above, what is the time of flight of this shot?

- . ---.--.--.-.. - --------- -------.--- ----.-.-------

(4) Calculate the horizontal distance that the shot travels after it leaves the athlete's hand.

Pm&ctile Motion

15 Two baseball players are throwing a ball to each other as shown at right. The ball is released and caught at the same height above ground level.

(1) Taking the upward direction to be positive, on the axes below, sketch graphs of the following

(a) the horizontal velocity of the ball whilst in flight;

(b) the acceleration of the ball whist in flight;

(c) the vertical velocity of the ball whilst in flight;

(d) the kinetic energy of the ball whilst in flight.

(2) Explain how each of the graphs would change if the ball was not caught by the player on the right, but was allowed to fall to the ground.

(a) horizontal velocity v,

lb) acceleration a

(c) vertical velocity v,

(d) kinetic energy K

20 PhysicsEssentials Stage 2 Physics

16 The diagram at the right shows the trajectory of a projectile C which is launched from the top of a building with an initial speed of vo at an angle of 8 above the horizontal. Five points are marked on the path of this projectile.

(1) What is the initial horizontal velocity of the projectile?

(2) What is the initial vertical velocity of the projectile?

-."" --------.--------------------"---.-..-.-- -.-" - .... ""

(3) At which of the marked points, if any, are the speeds of the projectile the same?

(4) At which of the marked points, if any, are the velocities of the projectile the same?

(5) At which of the marked points, if any, is the vertical velocity zero?

( 6 ) At which of the marked points, if any, is the horizontal velocity zero?

(7) What is the direction of the acceleration at points B and E?

(8) What are the directions of the velocity at points B and E?

- - -- - - --

(9) What is the magnitude of the vertical velocity and of the acceleration at point C?

17 A projectile is launched from ground level with a speed of u ms-I at an angle 8. Show that the range R u2 sin28

is given by R = , and that range is maximum when 8 = 45". (Note: 2 sin 8 cos 8 = sin 28 ) g

orm C ar Mo Syllabus Statement

In projectile motion, the example of motion in two dimensions introduced in Topic 1, the force and acceleration in the absence of air resistance are constant in both magnitude and direction. This second example of motion in two dimensions involves an object moving with constant speed in a circle (referred to as "uniform circular motion"). In uniform circular motion the force and acceleration continually change direction and are always directed towards the centre of the circle. The force is always perpendicular to the velocity. The resulting acceleration produces a continual change in the direction of the velocity without changing the magnitude of the velocity.

The theory is applied to the banking of road curves.

Key Ideas Students should know and understand the following

Intended Student Outcomes Students should be able to do the following

Centripetal Acceleration

1.1

1.2

1.3

1.4

1.1

1.2

1.3

The velocity of an object moving with uniform circular motion continually changes direction, and hence the object accelerates.

Average acceleration a for motion in more than one dimension is defined as a = Av /At where Av = vj- vi.. The acceleration a at any instant is obtained by allowing the time interval At to become very small.

The acceleration of an object moving with uniform circular motion is directed towards the centre of the circle and is called "centripetal acceleration".

The magnitude of the centripetal acceleration is constant for a given speed and radius and given by a = v2/r.

Using a vector argument, show that the change in velocity, and hence the acceleration, of an object over a very small time interval, is directed towards the centre of the circle.

Using the relationship v = 2nr/T, relate the speed v to the period T.

Solve problems involving the use of the equations a = v2/r and v = 2nrlT.

Force Causing the Centripetal Acceleration

2.1 A net force directed towards the centre of the circle is necessary to produce the centripetal acceleration.

Application: The Banking of Road Curves

2.1

3.1

3.2

3.3

3.4

Describe situations in which the centripetal acceleration is caused by a tension force, a frictional force, a gravitational force, or a normal force.

Identify the vertical and horizontal forces on a vehicle moving with constant velocity on a flat horizontal road.

Explain that when a vehicle travels round a banked curve at the correct speed for the banking angle, the horizontal component of the normal force on the vehicle (not the frictional force on the tyres) causes the centripetal acceleration.

Derive the equation tan0 = v2/rg, relating the banking angle 0 to the speed v of the vehicle and the radius of curvature r.

Solve problems involving the use of the equation tan0 = v2/rg.