Physics 160 Biomechanics

Projectiles

What is a Projectile?

• A body in free fall that is subject only to the forces of gravity and air resistance.

• Air resistance can often be ignored in shot-put, long jump and high jump but is a considerable factor in baseball, cycling, swimming and skydiving.

A human in flight is also a projectile.

Questions to think about

• If you want to throw a ball as far as possible what angle should you release it at?

• What can a diver do to increase the peak height of his dive?

• How can a gymnast increase her time in the air during a dismount?

• How can a punter increase both the distance and the hang time of his punts?

Projectile Motion • Many sports involve projectile motion.

• Objects acting as projectiles:

– Basketball, baseball, football, shot put, hammer, discus, javelin, golf ball, volleyball, tennis ball

• The body acts as a projectile in: – High jump, long jump, gymnastics, diving,

figure skating, ski jumping

• Variables of interest include: – Flight distance – Flight time – Maximum height

Kinematics of Projectile Motion

The horizontal and vertical components of projectile motion are analyzed separately

-the vertical component is influenced by gravity and the horizontal component is not

Kinematics of Projectile Motion

Two balls - one dropped and one projected horizontally from the same height:

Both land at the same time since gravity affects their vertical velocities equally.

Gravity and Horizontal Velocity

• Gravity does not change the horizontal velocity of an object.

( ) ( )horizontal initial horizontal finalv v=

vhorizontal

vvertical

a= - 9.8 m/s2

Influence of initial position on the shape of the trajectory

• projection velocity

• projection angle

• projection height.

Factors Influencing Projectile Trajectory

Factors Influencing Projectile Trajectory

Max

imum

hei

ght (

m)

Range (distance) (m) 0 1 2 3 4 5 6 7 8 9 10 11

5

4

3

2

1

0

This scaled diagram shows the size and shape of trajectories for an object projected at 10 m/s at different angles.

Optimum Angle of Projection

Factors Influencing Projectile Trajectory

FACTORS INFLUENCING PROJECTILE MOTION (Neglecting Air Resistance)

Variable Factors of Influence Flight time Initial vertical velocity

Relative projection height Horizontal displacement Horizontal velocity

Initial vertical velocity Relative projection height Vertical displacement Initial vertical velocity

Trajectory Initial speed Projection angle Relative projection height

Optimum Angle of Projection

• You need a large horizontal velocity, but if you sacrifice vertical velocity you have little time in the air

• Optimum angle without air resistance is 45o

Horizontal and Vertical Velocities *cos*sin

[ / ][ / ]

[ / ][deg]

x

y

x

y

v vv vv Horizontal velocity in m sv Vertical velocity in m sv Projection velocity in m s

Projection Angle in

θθ

θ

==

==

==

Range Equation

• Θ is the angle of projection • vx is the horizontal component of velocity • vy is the vertical component of velocity • h is the height of release of the projectile • g=9.8m/s2

2 2x y x yv v v v ghRange

g+ +

=

Example A shot putter releases

the shot at an angle of 40o from a height of 2.2 m with a velocity of 13.3 m/s. Find the range of the shot.

Ans. 20.1 m

Factors Influencing Projectile Trajectory

The Effect of Projection Angle on Range (Relative Projection Height = 0)

Projection Projection Speed Angle Range (m/s) (degrees) (m) 10 10 3.49 10 20 6.55 10 30 8.83 10 40 10.04 10 45 10.19 10 50 10.04 10 60 8.83 10 70 6.55 10 80 3.49

Time of flight for a projectile

2 2y y

x

v v ghtime

gRange v time

+ +=

= ×

Example

A baseball is thrown with a velocity of 31 m/s at an angle of 40o from a height of 1.8 m.

a) How long is it in the air? b) What is its range?

Maximum Height

• At the apex the vertical velocity is zero. • H is the maximum height above the release point.

( ) ( )2 2sin

2 2yv v

Hg g

θ= =

Example

A football is punted into the air. It has an initial vertical velocity of 15 m/s and an initial horizontal velocity of 16 m/s when it leaves the kicker’s foot. What is the maximum height reached by the football?

Example • A baseball leaves the bat at 44 m/s at an angle of 36o

from a height of 1.2 m above the ground . • A) Where does the ball hit the ground? • B) How long is the ball in the air? • C) What maximum height above ground does the ball

reach? • D) Which change increases the range of the ball more:

increasing the initial velocity to 48m/s leaving all other variables the same or increasing the angle of projection to 40o leaving all other variables the same?