A rock is thrown 28 degrees above the horizontal with an initial velocity of of at 8 meters per second off a 50m high cliff. Calculate how long it takes to hit the ground

Answer 1

To calculate the time it takes for the rock to hit the ground, we need to consider the vertical and horizontal components of the initial velocity separately.

Given:
- Initial velocity (vi) = 8 m/s
- Launch angle (θ) = 28 degrees
- Height of the cliff = 50m

Step 1: Calculate the vertical component of the initial velocity (viy) and acceleration due to gravity (g).
viy = vi * sin(θ)
viy = 8 * sin(28)
viy ≈ 3.63 m/s

Acceleration due to gravity, g = 9.81 m/s²

Step 2: Use the kinematic equation for vertical motion to calculate the time it takes for the rock to fall from the cliff to the ground. The equation is:
y = viy * t + 0.5 * g * t²

where:
- y = height of the cliff = 50m
- viy = vertical component of initial velocity = 3.63 m/s
- t = time taken to hit the ground

Substitute the values into the equation:
50 = 3.63 * t + 0.5 * 9.81 * t²

Rearrange the equation into a quadratic form:
0.5 * 9.81 * t² + 3.63 * t - 50 = 0

Solve the quadratic equation to find the time t. The solution for t should be the positive root, as time cannot be negative.

t ≈ 2.09 seconds

Therefore, it takes approximately 2.09 seconds for the rock to hit the ground.