You throw a rock straight up and find that it returns to your hand 3.40 s after it left your hand. Neglect air resistance. What was the maximum height above your hand that the rock reached?

The maximum height of the rock is 14.2 m

Explanation:

The equations that describe the height and velocity of the rock are the following:

y = y0 + v0 · t + 1/2 · g · t²

v = v0 + g · t

Where:

y = height of the object at time t

y0 = initial height

t = time

g = acceleration due to gravity (-9.8 m/s² if upward is positive)

v = velocity of the object at time t

We know that at t = 3.40 s, the rock is in your hand again. Then, if we place the origin of the frame of reference at your hand, the position of the rock at 3.40 s is 0 m. Using the equation of the position, we can calculate the initial velocity that we will need to obtain the max-height.

y = y0 + v0 · t + 1/2 · g · t²

0 = v0 · 3.40 s - 1/2 · 9.8 m/s² · (3.40 s) ²

(1/2 · 9.8 m/s² · (3.40 s) ²) / 3.40 s = v0

v0 = 16.7 m/s

At max-height, the velocity of the rock is 0. Then, using the equation of velocity we can calculate the time it takes the rock to reach the max-height. With that time, we can calculate the maximum height.

v = v0 + g · t (at max-height, v = 0)

0 = 16.7 m/s - 9.8 m/s² · t

- 16.7 m/s / - 9.8 m/s² = t

t = 1.70 s

Now, using this time in the equation of height:

y = y0 + v0 · t + 1/2 · g · t²

y = 0 m + 16.7 m/s · 1.70 s - 1/2 · 9.8 m/s² · (1.70 s) ²

y = 14.2 m

The maximum height of the rock is 14.2 m