Projectile Motion Chapter 5. Objectives Distinguish between a vector quantity and a scalar quantity....

Projectile MotionProjectile Motion

Chapter 5Chapter 5

ObjectivesObjectives Distinguish between a vector quantity and a Distinguish between a vector quantity and a

scalar quantity.scalar quantity. Explain how to find the resultant of 2 Explain how to find the resultant of 2

perpendicular vectors.perpendicular vectors. Describe how the components of a vector affect Describe how the components of a vector affect

each other.each other. Describe the components of projectile motion.Describe the components of projectile motion. Describe the downward motion of a horizontally Describe the downward motion of a horizontally

launched projectile.launched projectile. Describe how far below an imaginary straight-Describe how far below an imaginary straight-

line path a projectile falls. line path a projectile falls.

5.1 Vector and Scalar Quantities5.1 Vector and Scalar Quantities

Recall from Ch. 2Recall from Ch. 2A A vector quantityvector quantity is a quantity which is is a quantity which is

fully described by fully described by both magnitude and both magnitude and directiondirection. Velocity and acceleration are . Velocity and acceleration are vector quantities. vector quantities.

A A scalar quantityscalar quantity is a quantity which is is a quantity which is fully described by its fully described by its magnitudemagnitude. Speed . Speed is a scalar quantity.is a scalar quantity. Scalars can be added, subtracted, divided, and Scalars can be added, subtracted, divided, and

multiplied like any other number.multiplied like any other number.

5.2 Velocity Vectors5.2 Velocity Vectors Recall (from Ch. 2) that Recall (from Ch. 2) that

vectors have a head and vectors have a head and tail.tail.

When drawing a vector, a When drawing a vector, a scale is developed so scale is developed so that the magnitude of the that the magnitude of the vector in the diagram is vector in the diagram is represented by the length represented by the length of the arrow. of the arrow. The arrow is drawn a The arrow is drawn a

precise length in precise length in accordance with a chosen accordance with a chosen scale.scale.

5.2 Velocity Vectors5.2 Velocity Vectors

• The The directiondirection of a of a vector is expressed vector is expressed as the angle of as the angle of rotation of the rotation of the vector about its vector about its ““tailtail" from either " from either east, west, north, east, west, north, or south. or south.

• 3030oo west of north west of north

N

E

S

W


Velocity of something is often the result of Velocity of something is often the result of combining 2 or more velocities.combining 2 or more velocities.An airplane usually encounters a An airplane usually encounters a wind wind

(air which is moving with respect to an (air which is moving with respect to an observer on the ground below). observer on the ground below).

A motor boat in a river encounters a A motor boat in a river encounters a river river currentcurrent (water which is moving (water which is moving with respect to an observer on dry land).with respect to an observer on dry land).


The vectors are The vectors are scaled to represent scaled to represent 100 km/hr and 25 100 km/hr and 25 km/hr.km/hr.

Since these vectors Since these vectors are in the same are in the same direction, they are direction, they are “in parallel”.“in parallel”.

Recall (from Ch. 2) Recall (from Ch. 2) that if vectors are in that if vectors are in parallel, the parallel, the magnitude values magnitude values ADD.ADD.


The velocity The velocity vectors are now vectors are now in opposite in opposite directions.directions.

The magnitude The magnitude values will, values will, therefore, therefore, SUBTRACT.SUBTRACT.

5.2 Velocity Vectors5.2 Velocity Vectors Consider now the case Consider now the case

of the crosswind (or of the crosswind (or cross current).cross current).

The resultant (the The resultant (the result of adding these result of adding these 2 vectors) is the 2 vectors) is the DIAGONAL of the DIAGONAL of the rectangle created by rectangle created by the 2 vectors. the 2 vectors. (Remember the (Remember the “Pythagorean “Pythagorean Theorem” ?)Theorem” ?)


Since the two vectors to be added - Since the two vectors to be added - the southward plane velocity and the the southward plane velocity and the westward wind velocity - are at right westward wind velocity - are at right angles to each other, the angles to each other, the Pythagorean theorem can be used to can be used to determine the magnitude of the determine the magnitude of the resultant velocity.resultant velocity.

252522 + 100 + 10022 = r = r22

r = r = √√10625 = 103.1 km/hr.10625 = 103.1 km/hr.

PracticePracticeHow fast will a boat move upstream if it has How fast will a boat move upstream if it has

a speed of 20 mph and the current a speed of 20 mph and the current downstream is 5 mph?downstream is 5 mph?

How fast will the boat move directly across How fast will the boat move directly across stream if its speed is 16 mph and the stream if its speed is 16 mph and the current downstream is 12 mph?current downstream is 12 mph?

Suppose an airplane flying at 80 km/hr Suppose an airplane flying at 80 km/hr encounters a crosswind of 15 km/hr. Will encounters a crosswind of 15 km/hr. Will the airplane fly faster or slower than 80 the airplane fly faster or slower than 80 km/hr?km/hr?

5.3 Components of Vectors5.3 Components of Vectors A single vector will often need to be transformed A single vector will often need to be transformed

into 2 vectors that are at right angles to each into 2 vectors that are at right angles to each other.other.

The 2 vectors at right angles that add up to the The 2 vectors at right angles that add up to the given vector are known as the given vector are known as the components components of of the vector. the vector. The perpendicular components of The perpendicular components of a vector are independent of each other.a vector are independent of each other.

The process of determining the components of a The process of determining the components of a vector is known as vector is known as resolution.resolution.

There are 2 methods of vector resolution - a There are 2 methods of vector resolution - a graphical method and a trigonometric method.graphical method and a trigonometric method.

Problem: Graphical Vector Problem: Graphical Vector ResolutionResolution

If Fido's dog chain is If Fido's dog chain is stretched upward stretched upward and rightward and and rightward and pulled tight by his pulled tight by his master, then the master, then the force in the chain force in the chain has two has two components - an components - an upward and a upward and a rightward rightward component.component.

GraphicalGraphical Vector Resolution Vector Resolution First, using a scale, draw the First, using a scale, draw the

vector you want to resolve.vector you want to resolve. Next, draw in x and y axes at the Next, draw in x and y axes at the

tail of the vector you want to tail of the vector you want to resolve.resolve.

For the rightward component, For the rightward component, draw a dotted line from the end draw a dotted line from the end of the vector, parallel to the y-of the vector, parallel to the y-axis, to its intersect on the x axis. axis, to its intersect on the x axis.

Indicate the rightward Indicate the rightward component by drawing a vector component by drawing a vector from the origin to the dotted line.from the origin to the dotted line.

Vector representing force on Fido

Rightward component

GraphicalGraphical Vector Resolution Vector Resolution

For the upward component, For the upward component, draw a dotted line from the end draw a dotted line from the end of the vector, parallel to the x-of the vector, parallel to the x-axis, to its intersect on the y axis, to its intersect on the y axis. axis.

Indicate the upward Indicate the upward component by drawing a vector component by drawing a vector from the origin to the dotted from the origin to the dotted line.line.

Measure (using your scale) Measure (using your scale) both components.both components.

Upward component

PracticePractice Graphically Graphically

resolve a resolve a velocity vector of velocity vector of 100 km/hr that 100 km/hr that is at an angle of is at an angle of 606000 to the x-axis. to the x-axis.

Graphically Graphically resolve a force resolve a force vector of 350 N vector of 350 N that is at an that is at an angle of 25 angle of 25 00to to the x-axis.the x-axis.

Problem: Trigonometric Vector Problem: Trigonometric Vector ResolutionResolution

1.1. Construct a sketch (no Construct a sketch (no scale needed) of the scale needed) of the vector in the indicated vector in the indicated direction; label its direction; label its magnitude and the magnitude and the angle which it makes angle which it makes with the horizontal. with the horizontal.

2.2. Draw a rectangle about Draw a rectangle about the vector such that the the vector such that the vector is the diagonal of vector is the diagonal of the rectangle; draw the the rectangle; draw the components of the components of the vector and label them. vector and label them.

Trigonometric Vector ResolutionTrigonometric Vector Resolution

3. a. With the given vector (c in the diagram below) 3. a. With the given vector (c in the diagram below) and angle (and angle (θθ below) below), use the sine function to , use the sine function to solve for the x component (b in diagram).solve for the x component (b in diagram).

b. Use the cosine function to solve for the y b. Use the cosine function to solve for the y component (a in diagram).component (a in diagram).

Trigonometric Vector ResolutionTrigonometric Vector Resolution

Fvert

Fhoriz

PracticePractice

A ball is thrown upwards at an angle of 60A ball is thrown upwards at an angle of 6000 from from the ground. If its initial velocity is 40 m/s, what is the ground. If its initial velocity is 40 m/s, what is the horizontal component of the velocity? The the horizontal component of the velocity? The vertical component?vertical component?

Carrie digs a hole in her backyard to plant the Carrie digs a hole in her backyard to plant the rose bush her kids gave her for Mother’s Day. If rose bush her kids gave her for Mother’s Day. If she exerts a force of 30. N on the shovel at an she exerts a force of 30. N on the shovel at an angle of 40angle of 4000to the ground, what are the to the ground, what are the horizontal and vertical components of Carrie’s horizontal and vertical components of Carrie’s force?force?

5.4 Projectile Motion5.4 Projectile Motion

A projectile is any object that moves through air A projectile is any object that moves through air or space acted on only by gravity and air or space acted on only by gravity and air resistance.resistance.


There are two There are two components of any components of any projectile's motion – projectile's motion – the horizontal the horizontal component and the component and the vertical component. vertical component.

Horizontal ComponentHorizontal Component

The horizontal The horizontal component is just like component is just like the horizontal motion the horizontal motion of a ball rolling freely of a ball rolling freely along a level surface along a level surface with no friction.with no friction.

With no friction, the With no friction, the horizontal velocity is horizontal velocity is constant. (v = d/t) constant. (v = d/t)

With no outside With no outside forces, the horizontal forces, the horizontal acceleration is 0.acceleration is 0.

Vertical ComponentVertical Component

The vertical component is The vertical component is like the motion of a freely like the motion of a freely falling object.falling object.

A projectile’s velocity A projectile’s velocity increases as it falls - it increases as it falls - it accelerates at about 10 accelerates at about 10 m/sm/s2.2. (v = gt) (v = gt)

In each time interval, the In each time interval, the projectile covers a greater projectile covers a greater distance. (d = 1/2gtdistance. (d = 1/2gt22))


The horizontal and vertical components of the The horizontal and vertical components of the motion are INDEPENDENT of each other.motion are INDEPENDENT of each other.

Practice ProblemPractice Problem

Gerrie rolls a marble off a table. If the Gerrie rolls a marble off a table. If the marble hits the floor at a distance that is marble hits the floor at a distance that is 0.40 m from the table’s edge exactly 0.50 0.40 m from the table’s edge exactly 0.50 seconds later, what is . . .seconds later, what is . . .

Horizontal component of the marble’s Horizontal component of the marble’s velocity?velocity?

The vertical velocity at touchdown?The vertical velocity at touchdown?The height of the table?The height of the table?

5.5 Projectiles Launched 5.5 Projectiles Launched HorizontallyHorizontally


Notice (in the previous slide) that the Notice (in the previous slide) that the horizontal motion remains constant. The horizontal motion remains constant. The canon ball moves the same distance in canon ball moves the same distance in each second because there are no forces each second because there are no forces acting on it.acting on it.

Gravity acts downward so the only Gravity acts downward so the only acceleration that occurs is downward. The acceleration that occurs is downward. The downward motion of a “launched” downward motion of a “launched” projectile is the SAME as that of free fall.projectile is the SAME as that of free fall.


The path traced by a The path traced by a projectile is a projectile is a parabola; parabola; the curved the curved path is considered path is considered parabolicparabolic..

The trajectory (path) The trajectory (path) is a combination of is a combination of constantconstant horizontal horizontal motion and motion and acceleratedaccelerated vertical vertical motion.motion.

Practice ProblemPractice Problem George is working on his house. From a ladder, George is working on his house. From a ladder,

he tosses his hammer to the ground, 2.8 m he tosses his hammer to the ground, 2.8 m below his position. The hammer lands 1.5 m below his position. The hammer lands 1.5 m horizontally from George.horizontally from George.

How long is the hammer in the air?How long is the hammer in the air? What is the horizontal velocity component of What is the horizontal velocity component of

George’s throw?George’s throw? If George’s initial throw has no vertical If George’s initial throw has no vertical

component, what is the vertical velocity of the component, what is the vertical velocity of the hammer when it hits the ground?hammer when it hits the ground?

What is the velocity of the hammer when it strikes the What is the velocity of the hammer when it strikes the ground?ground?

5.6 Projectiles Launched at an 5.6 Projectiles Launched at an AngleAngle


This diagram shows that This diagram shows that there are specific vertical there are specific vertical distances for the cannonball distances for the cannonball from the position of where it from the position of where it should be (with no gravity) to should be (with no gravity) to where it actually is (with where it actually is (with gravity).gravity).

These distances are the These distances are the SAME distances it would fall SAME distances it would fall if it were dropped and were if it were dropped and were undergoing free fall for the undergoing free fall for the given amount of time.given amount of time.

1 s 5 m 20m 45m 80m 125m

2 s3 s

4 s

5 s


Recall (from Ch. 4) that the distance is Recall (from Ch. 4) that the distance is given by d = 1/2atgiven by d = 1/2at22..

Since a = g = 9.8 m/sSince a = g = 9.8 m/s22, d = 4.9t, d = 4.9t22..No matter the angle at which a projectile is No matter the angle at which a projectile is

launched, launched, the vertical distance of fall from the vertical distance of fall from an idealized straight-line path will always an idealized straight-line path will always be the same for equal times.be the same for equal times.

Practice ProblemPractice Problem

Mario, the amazing human cannonball, is Mario, the amazing human cannonball, is fired out of a cannon at a speed of 20. m/s fired out of a cannon at a speed of 20. m/s at an angle of 43at an angle of 4300 to the horizontal. to the horizontal.

What are the horizontal and vertical What are the horizontal and vertical components of his initial speed?components of his initial speed?

If he lands 52 m away from the canon, If he lands 52 m away from the canon, how long was he in the air?how long was he in the air?

Assuming the canon is at ground level, Assuming the canon is at ground level, what vertical height did Mario achieve?what vertical height did Mario achieve?

HeightHeight

The vertical distance The vertical distance a projectile falls below a projectile falls below an imaginary straight-an imaginary straight-line path increases line path increases continually with time continually with time and is equal to 4.9tand is equal to 4.9t22 meters.meters.

HeightHeight

The diagram above shows the path of a projectile with The diagram above shows the path of a projectile with velocity vectors drawn in for different points on its velocity vectors drawn in for different points on its parabolic path. The vectors have been resolved into their parabolic path. The vectors have been resolved into their x and y components.x and y components.

HeightHeight

The horizontal component is always the same – the ball The horizontal component is always the same – the ball moves equal distances in equal time intervals.moves equal distances in equal time intervals.

Note that ONLY the vertical component changes – it Note that ONLY the vertical component changes – it decreases going upward against gravity and increases decreases going upward against gravity and increases going downward with gravity. The projectile accelerates going downward with gravity. The projectile accelerates due to Earth’s gravity.due to Earth’s gravity.

Is the velocity of the projectile at its highest point zero?Is the velocity of the projectile at its highest point zero?

RangeRange

Now suppose 2 projectiles are Now suppose 2 projectiles are launched at the same speed but launched at the same speed but different angles. The projectile different angles. The projectile launched at the greater angle will launched at the greater angle will have a velocity vector with a have a velocity vector with a greater vertical component and a greater vertical component and a smaller horizontal component.smaller horizontal component.

The greater vertical component The greater vertical component results in a higher path; the results in a higher path; the smaller horizontal component smaller horizontal component results in less range (distance).results in less range (distance).

Horizontal RangesHorizontal Ranges

Here the paths of 3 Here the paths of 3 cannonballs are shown – cannonballs are shown – all launched at the same all launched at the same initial speed but different initial speed but different projection angles.projection angles.

Note:Note: The paths are all The paths are all

parabolas.parabolas. Each ball reaches a Each ball reaches a

different height.different height.

What’s going on with the What’s going on with the horizontal ranges?horizontal ranges?

Horizontal RangesHorizontal Ranges

Different projection angles – 30Different projection angles – 3000 and 60 and 6000, for example – , for example – can have the same range. This always works out for can have the same range. This always works out for angles that add up to 90angles that add up to 9000..

For angles less than 45For angles less than 4500, the smaller the angle, the , the smaller the angle, the shorter the horizontal range.shorter the horizontal range.

Maximum range is usually attained at an angle of 45Maximum range is usually attained at an angle of 4500. .

SpeedSpeed

The cases of projectile motion mentioned in The cases of projectile motion mentioned in class assume air resistance is negligible.class assume air resistance is negligible.The speed lost by the projectile going up equals The speed lost by the projectile going up equals

the speed gained while it is coming down.the speed gained while it is coming down.The time to go up equals the time to come The time to go up equals the time to come

down.down.With air resistance, the range of the With air resistance, the range of the

projectile is diminished and the path is not a projectile is diminished and the path is not a true parabola.true parabola.

SpeedSpeed

Also, the projectile motion discussed in class Also, the projectile motion discussed in class has been “short-range” projectile motion. For has been “short-range” projectile motion. For this motion, we assume the ground is flat - the this motion, we assume the ground is flat - the curvature of the Earth’s surface has no effect on curvature of the Earth’s surface has no effect on the motion.the motion.

For “long-range” projectiles, the curvature of the For “long-range” projectiles, the curvature of the Earth must be taken into account. If a projectile Earth must be taken into account. If a projectile has enough initial velocity, it will “fall all the way has enough initial velocity, it will “fall all the way around the Earth” and become a satellite. around the Earth” and become a satellite.

Projectile Motion Chapter 5. Objectives Distinguish between a vector quantity and a scalar quantity....

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