A home run is hit in such a way that the baseball just clears a wall 18.0 m high

Question

A home run is hit in such a way that the baseball just clears a wall 18.0 m high, located 132 m from home plate. The ball is hit at an angle of 33.0° to the horizontal, and air resistance is negligible. (Assume that the ball is hit at a height of 1.0 m above the ground.)

(a) Find the initial speed of the ball.
m/s

(b) Find the time it takes the ball to reach the wall.
s

(c) Find the velocity components of the ball when it reaches the wall.

Find the speed of the ball when it reaches the wall.
m/s

Explanation / Answer

The ball is hit 33° to the horizontal so the relation between the x and y components of the initial velocity is:
tan 33° = vy / vx
vy = vx * tan 33°

vx is constant, but vy changes with time.

x = 132m
y = 17m (21m - 1m)

Therefore:
x = vx*t
y = vy*t +1/2*g*t²

Plug in known values:
132m = vx*t
vx = 132m / t

Substitute for vy then vx:
17m = vx * tan 33° * t + 1/2 * -9.81m/s² * t²
17m = 132m / t * tan 33° * t + 1/2 * -9.81m/s² * t²

The t cancel out:
17m - 132m*tan 33° = 1/2 * -9.81m/s² * t²
t = ((17m - 132m*tan 33°) / (1/2 * -9.81m/s²))
t = 3.743 s

The ball travels 3.743 s before going over the fence

a)
We can use the constant x component of the velocity to calculate the initial velocity.
The relation between the velocity and its x component is:
cos 33° = vx / velocity

vx = dx / t = 132m / 3.743 s = 35.265 m/s

velocity = vx / cos 33° = 35.265 m/s / cos 33° = 42.0487 m/s

b)
Already calculated: t = 3.743 s

c) vx is constant at 35.265 m/s

vy can be calculated using the equation:
v = v0 + a*t

Initial vy = 42.0487 m/s * sin 33° = 22.9 m/s
Final vy = 22.9 m/s + (-9.81m/s²) * 3.743 s Final vy = -13.81883 m/s

d) The speed when the ball reaches the wall is (using Pythagorean Theorem):
v = (vx² + vy²) = ((35.265 m/s)² + (-13.81883 m/s)²)
v = 37.8759 m/s

Answers:
a) Initial ball speed: 42.0487 m/s
b) Time for ball to reach wall: 3.743 s
c) x velocity-component: 35.265 m/s
y velocity-component: -13.81883 m/s
d) Speed of the ball: 37.8759 m/s

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