Projectile MotionProjectile Motion

Projectile VelocitiesProjectile Velocities

Sammy ACT:Sammy ACT: Sammy Sosa clobbers a fastball toward

center-field. The ball is hit 1 m (yo ) above the plate, and its initial velocity is 36.5 m/s (vo ) at an angle of 30o () above horizontal. The center-field wall is 113 m (x) from the plate and is 3 m (h) high.What time does the ball reach the fence?What time does the ball reach the fence?Does Sammy get a home run?Does Sammy get a home run?

In other words, will the ball clear the fence?In other words, will the ball clear the fence?

vvoo

h

x

y0

Sammy ACT:Sammy ACT:

vvoo

y

x

y0

Choose y axis up. Choose x axis along the ground in the

direction of the hit. Choose the origin (0,0) to be at the plate. Say that the ball is hit at t = 0, x = x0 = 0

y

x

Sammy ACTSammy ACTEquations of motion are:

Horizontal Vertical

vx = constant vy = vy0 - gt

x = vxt y = y0 + vy0 t - 1/ 2 gt2

Remember that the horizontal and vertical motions are Independent of one another, but they share the time.

Variables: v0 = 36.5 m/s t = ???

= 30o is y > h = 3 m??? y0 = 1 m

x = 113 m

Sammy...Sammy...

Use geometry and trigonometry to figure out vx and vy0 :

y

x

g

vv00

vx

vy0

Find vx = v cos .

and vy = v sin .

y0vx = 36.5 cos(30o) m/s = 31.6 m/s

vy0 = 36.5 sin(30o) m/s = 18.25 m/s

36.5 m/s

300

Sammy ACTSammy ACT We can use the horizontal motion equation to

determine the time to reach the wall: x = vx t t = x / vx t = (113 m) / (31.6 m/s) = 3.58 s We have an equation that tells us

y(t) = y0 + vy0 t + a t2/ 2

y(t) = (1.0 m) + (18.25 m/s)(3.58 s) - (0.5)(9.8 m/s2)(3.58 s)2

= (1.0 + 65.3 - 62.8) m = 3.5 mm

Since the wall is 3 m high, Sammy gets the homer!!

Projectile RangeProjectile Range

Equations of motion are: Horizontal Vertical

vx = constant vy = vy0 - gt

x = vxt y = y0 + vy0 t - 1/ 2 gt2

Equations for x and y can be combined (click here to see the derivation) to yield the following equation: 2

0max

sin(2 )vx

g

Range of ProjectileRange of Projectile

Golf Range ACTGolf Range ACT A golf ball is chipped at an angle of 46o and

with a speed of 5.4 m/s. How far does it travel to the nearest tenth of a meter?

20

max

sin(2 )vx

g

2

2

max

(5.4 ) sin(2 46 )

9.8

oms

ms

x

max 2.97x m

Projectile Launched at AngleProjectile Launched at Angle

Feeding the MonkeyFeeding the Monkey(Banana Gun)(Banana Gun)

Where does the zookeeper aim if he wants to hit the monkey?

( He knows the monkey willlet go as soon as he shoots ! )

Feeding the Monkey...Feeding the Monkey... If there were no gravity, simply aim

at the monkey

r = r0

r =v0t

Feeding the Monkey...Feeding the Monkey...

rr = vv0 t - 1/2 gg t2

With gravity, still aim at the monkey! rr = r0 - 1/2 gg t2

Banana hits

monkey!