PROJECTILE MOTIONCourse: Diploma

Subject: Applied Science Physics

Unit: IV

Chapter: I

Projectile motion

Projectile is a body thrown with an initial velocity in the vertical plane and then it moves in two dimensions under the action of gravity alone without being propelled by any engine or fuel. Its motion is called projectile motion.

The path of a projectile is called its trajectory.

Examples:

1. A packet released from an airplane in flight.

2. A golf ball in flight.

3. A bullet fired from a rifle.

4. A jet of water from a hole near the bottom of a water tank.

If gravity were not present, a projectile would travel in a constant straight line.

However, the presence of gravity forces projectiles to travel in a parabolic trajectory, thus gravity accelerates objects downwards.

The factors that affect the trajectory are: a) Angle of projection b) Projection velocity c) Relative height of projection

Projection speed

Projection height

Projection angle

Factors Affecting the Trajectory of a Projectile

Optimum Projectile Angle or Angle of projectionThe angle at the projectile body is thrown

is called Optimum Projectile Angle or Angle of projection

Optimum Projectile Angle or Angle of projection

If Relative Projection Height = 0, the optimum angle = 450

If Relative Projection Height > 0 , the optimum angle < 450

If Relative Projection Height < 0 , the optimum angle > 450

The time of flight

The time of flight of a projectile motion is the time from when the object is projected to the time it reaches the surface. As we discussed previously, T depends on the initial velocity magnitude and the angle of the projectile:

yuT

g

u sinT

g

Maximum height

maximum height, which is the height of the projectile when the vertical velocity is zero and the projectile has only its horizontal-velocity component.

2y

2

max

max

y

2 2 2 2

2

2 2

u sinT & u u sin

g

2u sin

g

u sin

1X u t at

2

X h & a g & t

u sin 1h u

u

sin gg 2

1h g

2

h

sin

g g

u sin

2g

Horizontal rangeThe range, R, which is the horizontal distance

that the projectile travels.Range RRange is the total horizontal distance covered during the time of flight.From equation for horizontal motion, x=uxt When t=T , x=RR= uxT = ucos.2usinΦ/g= u22sinΦcosΦ/g = u2sin2Φ/g using 2sinΦcosΦ= sin2ΦR= (u2sin2Φ)/g

Equation of Trajectory (Path of projectile)

At any instant tx= u.cosΦ.tt= x/(u.cosΦ)

Also , y= u sinΦ.t - (1/2)gt2

Substituting for ty= u.sinΦ.x/(u.cosΦ) - (1/2)g[x/(u.cosΦ)]2

y= x.tanΦ - [(1/2)g.sec2.x2 ]/u2

This equation is of the form y= ax + bx2 where 'a' and 'b are constants. This is the equation of a parabola.

Thus, the path of a projectile is a parabola .

A lava rock is thrown horizontally at 15m/s from the top of a cliff 44 m high.

a. How far form the base of the cliff does the rock hit the ground?

KNOWN:

xy = -44 m xx =?

ay= -9.8 m/s/s ax = 0

vi,y = 0 vi,x = 15m/s

vf,x = 0

First we need to solve for the time.

Use the formula xy = vit + 1/2at2 Time = 3 secondsNow we know the time and we can find how far

from the base of the cliff it landed.Use the formula v= x/t and the distance is 45 m

Answer: 40 m/s

The movie “The Gods Must Be Crazy” begins with a pilot dropping a bottle out of an airplane.

It is recovered by a surprised native below, who thinks it is a message from the gods.

If the plane was flying at an altitude of 500 m, and the bottle lands 400 m horizontally from the initial dropping point, how fast was the plane flying.

Answer : B

An autographed baseball rolls off of a 0.70m high desk and strikes the floor 0.25 m away from the base of the desk. How fast was it rolling?

A. 0.77 m/sB. 0.66 m/sC. 0.54 m/s

Answer : A

A projectile is…

A. An object thrown or launched into the air.    

B. An object rolling down a hill

C. An object that is being hit by a bat or other object

    

Answer : C

A cat chases a mouse across a 1.0 m high table. The mouse steps out of the way, and the cat slides off the table and strikes the floor 2.2m from the edge of the table. What was the cat’s speed when it slid off the table?

A. 5.9 m/s

B. 2.4 m/s

C. 4.9 m/s

Answer : B

A pelican flying along a horizontal path drops a fish from a height of 5.4 m. The fish travels 8.0 m horizontally before it hits the water below. What is the pelican’s initial speed?

A. 5.4 m/sB. 7.7 m/sC. 9.8 m/s

Answer : B

The path a projectile takes is called_________

A. Parabola

B. Trajectory

C. equilibrium

    

Answer : A

During a thunderstorm, a tornado lifts a cow to a height of 125 above the ground. Increasing in strength, the tornado flings the cow horizontally with an initial speed of 90 m/s. How long does the cow take to reach the ground? How far does the cow travel before hitting the ground?

A. 5.05 s ; 454mB 5.05 s ; 234 mC. 4.03 s ; 454 mD. 4.03 s ; 454 m

Answer: 22 m/s

A ball is thrown horizontally from the top of a building from a height of 32.5 meters above the ground. It lands 56 meters from the base of the building. What is the initial velocity of the ball? ( Two significant digits)

Answer : C

In the x direction (ignoring air resistance), the force acting on the projectile is________

A. The force that launched

B. Gravity

C. There are no forces in the x direction    

Answer : A

I

n

t

h

e

x

d

irecti

o

n

(i

g

n

o

ri

n

g

air

resista

n

ce),

t

h

e

accelerati

o

n

o

f

t

h

e

p

r

o

jectile

is

_

_

_

_

A. Zero

B. 9.8 m/s

C. Increasing by 9.8 m/s every second.   

Answer : c

In the x direction (ignoring air

resistance), the velocity of a

projectile______

A. Increases

B. Decreases

C. Stays the same

Answer : C

In the x direction, the object’s speed ______. (Ignoring air resistance)\

A. Increases

B. Decreases

C. Stays the same

Answer : A

A projectile is an object that the only forces acting on it are air resistance and gravity.

A. True

B. False

Answer : B

In the y direction (ignoring air resistance), the force acting on the

projectile is ______

A. The force that launched it

B. Gravity

C. There is no force in the Y direction

Answer: B

If an object is shot horizontally, the initial velocity in the y direction is_____________

A. 9.8 m/s

B. Zero

C. Dependent on the time

Answer: C

The organs fly off the edge of a table with a velocity of ____m/s. The height of the table is 3 m and the horizontal range of the organs from the base of the table is 15 m. How long does it take for the organs to hit the ground?

A. .61sB. 9.8 sC. .78 sD. .57 s

Answer : 145.5 m

A bridge is 180 m above a river. If a goat is thrown from the bridge by a troll with a horizontal velocity of 24 m/s. How far has the goat moved horizontally when it hits the water?

Answer : D

A ball rolls off the edge of a table with a horizontal velocity of ___ m/s. The height of the table is 7 m and the horizontal range of the ball from the base of the table is 15 m. What is the initial speed of the ball.

A. 35 m/sB. 24 m./sC. 32 m/sD. 13 m/s

Answer : 14.6 m

Zeus throws a lightning bolt with a horizontal velocity of 25 m/s. The lightning bolt lands 30 meters away from the cloud. How high is the cloud?

A ball is launched with an initial velocity of 4.47 m/s at an angle of 66 degrees with the horizontal.

a. What was the maximum height the ball attained?

b. How long did it take the ball to return to the launching height?

c. What was its range?

0.86 m

0.84 s

1.5 meters

A rugby player kicks a ball from ground level with an initial velocity of 27.0 m/s at 30 degrees above the horizontal.

What is the….A. Hang timeB. rangeC. Maximum height

2.76 s

64.6 m9.3 m

Answer: 207 m

A can is kicked with an initial speed of 60 m/s at an angle of 30 degrees above the horizontal. What s the can’s horizontal displacement at the end of 4 seconds?

Answer : C

A baseball is hit with a vertical speed of 20 m/s and a horizontal speed of 30 m/s. How long will the ball remain in the air?

A. 2 secondsB. 3 secondsC. 4 secondsD. there is no way to tell

Answer: B

A girl kicks a robot and the head flies off at an angle of 20 degrees. The head would have landed at the same spot if it had been kicked at an angle of:

A: 40 degrees

b. 70 degrees

C. 45 Degrees

D. 25 degrees

Answer: 25 m/s

The horizontal and vertical components of the initial velocity of a football are 24 m/s and 7 m/s respectively. What is the initial velocity of the football?

Answer: 4.7 m

A bride throws a bouquet with an initial velocity of 27 m/s at an angle of 25 degrees above the horizontal. How high above the projection point is the bouquet after 1.8 s?

Answer : CA ball is thrown at an angle of 45

degrees above the horizontal with an initial velocity of 20m/s. What is the total time of flight of the ball before it hits the ground?

A. 9.8 sB. 6.1 sC. 2.9 sD. 4.9 s

Answer : D

A man kicks a soccer ball with an initial velocity of 25 m/s at an angle of 30°above the horizontal. The horizontal component of the velocity of the ball is described by one of the following statements.

A) It changes uniformly. B) It is zero. C) It is 25 m/s. D) It remains constant during the ball's

flight.

Answer : C

A man is launched from a cannon with an initial velocity of 26 m/s at an angle of 20 degrees above the horizontal. How high above the projection point is the man after 1.4 s?

A. 8.2 mB. 24 mC. 2.8 m

Answer: 469.5 meters

Suppose that an angel flying 60 m/s, at a height of 300m, dropped a sack of halos. How far from the point of release would the sack have traveled when it struck the ground?

A snowball is thrown at some angle above the horizontal with a certain velocity. It reaches its highest point and starts falling down. What is the velocity of the snowball at the highest point of the trajectory?

A. 9.8 m/sB. It is equal to its initial

horizontal velocityC. 0D. It is equal to its initial vertical

velocity.

Answer : B

Answer : A

Which of the following conditions is met when a projectile reaches its maximum height?

A. Vertical component of the velocity is zero.

B. Vertical component of the velocity is maximum.

C. Horizontal component of the velocity is maximum.

D. Acceleration in the vertical direction is zero.

Answer : C

What is the range of a projectile?

A. The total trajectory that the projectile travels.

B. The vertical distance that the projectile travels.

C. The horizontal distance that the object travels.

D. Twice the maximum height of the projectile.

Answer : B

The horizontal and vertical components of the initial velocity of a soccer ball are 24m/s and 7 m/s respectively. What is the initial velocity of the soccer ball?

A) 31 m/s B) 25 m/s C) 17 m/s D) 42 m/s

Answer : C

Define the flight time or hang time of a trajectory.

A. Time taken by the projectile to reach the maximum height.

B. The maximum height reached by the projectile divided by the magnitude of the vertical velocity.

C. The total time the projectile was in the air.

D. Half the total time the projectile was in the air.

Answer : B

. What is the maximum height obtained by a 125g rock that is slung from a slingshot at an angle of 78° from the horizontal with an initial velocity of 18 m/s?

A. 0.70 m

B. 16 m

C. 32 m

D. 33 m

Answer : D

An orange is dropped at the same time a bullet is shot from a gun ( ignoring air resistance). Which of the following is true?A. The acceleration due to gravity is greater for the

orange because the orange is heavier.

B. Gravity acts less on the bullet than on the orange because the bullet is moving so fast.

C. The velocities will be the same.

D. The two objects will hit the ground at the same time.

Answer : C

A baseball player throws a ball from left field toward the home plate. ( Neglecting air resistance) At the instance the ball reaches its highest point, what is the direction of the ball’s velocity?

A. UpB. DownC. horizontalD. There is not enough information to say

Answer : C

If an object is shot upward at an angle (no air) and lands at the same height it is shot, the final velocity is 

A. Zero

B. The same as the initial velocity

C. Negative the initial velocity

REFERENCE BOOKS AUTHOR/PUBLICATION

ENGINEERING PHYSICSB.L.THERAJA, S. CHAND

PUBLISHERS

1. http://exbiomech.tamu.edu/afp_hlknwebsrv.tamu.edu/KINE_426_files/11.PROJ.K426a.pptx

2. http://images.tutorvista.com/content/gravitation/projectile-flight-time.gif

3. http://3.bp.blogspot.com/_teko96dlZB4/TNbFs9tF4qI/AAAAAAAAACc/1JTTisEAqcM/s1600/projectile_motion_picture.jpg

4. https://figures.boundless.com/16970/full/range.gif

5. http://dallaswinwin.com/Motion_in_Two_Dimensions/projec1.gif