projectile motion projectile velocities sammy act: sammy sosa clobbers a fastball toward center-...
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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
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
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