A ball is launched off a ledge with an initial velocity of 14.0 m/s [E]. When it hits the ground, the angle of elevation from its landing point to the original launch position is 60 degrees. What are the vertical and horizontal displacements the ball?

Question

A ball is launched off a ledge with an initial velocity of 14.0 m/s [E]. When it hits the ground, the angle of elevation from its landing point to the original launch position is 60 degrees. What are the vertical and horizontal displacements the ball?

Thanks for any help! I've been stuck on this practice problem for a while now!

Answer 1

I assume that "14.0 m/s [E]" means the ball was launched horizontally.

So, if the initial height is h, then the distance traveled horizontally is h cos60° = h/2
the time taken to fall is t where
4.9t^2 = h
14t = h/2
so
4.9t^2 = 28t
t = 5.71 seconds
Now you can find h and h/2

Answer 2

Thanks so much oobleck! That really helped.

Answer 3

To find the vertical and horizontal displacements of the ball, we can break down the initial velocity into its vertical and horizontal components.

Given:
Initial velocity (v) = 14.0 m/s [E]
Angle (θ) = 60 degrees

Step 1: Calculate the vertical component of the initial velocity.
The vertical component (v_y) can be calculated using the formula:
v_y = v * sin(θ)

v_y = 14.0 * sin(60)
v_y ≈ 12.124 m/s

Step 2: Calculate the time (t) taken to reach the ground.
We can use the vertical component (v_y) to find the time it takes for the ball to land. Assuming there is no air resistance, the time of flight can be calculated using the following formula:
t = (2 * v_y) / g

where g is the acceleration due to gravity (approximately 9.8 m/s^2).

t = (2 * 12.124) / 9.8
t ≈ 2.47 s

Step 3: Calculate the vertical displacement (y).
The vertical displacement is given by the formula:
y = v_y * t + (1/2) * g * t^2

y = 12.124 * 2.47 + 0.5 * 9.8 * (2.47)^2
y ≈ 14.87 m

Therefore, the vertical displacement of the ball is approximately 14.87 meters.

Step 4: Calculate the horizontal displacement (x).
The horizontal displacement (x) can be calculated using the horizontal component of the initial velocity (v_x) and the time of flight (t). Since there is no acceleration in the horizontal direction, the horizontal velocity remains constant.

v_x = v * cos(θ)

v_x = 14.0 * cos(60)
v_x ≈ 7.0 m/s

x = v_x * t
x = 7.0 * 2.47
x ≈ 17.29 m

Therefore, the horizontal displacement of the ball is approximately 17.29 meters.

To summarize:
Vertical displacement (y): approximately 14.87 meters
Horizontal displacement (x): approximately 17.29 meters