Question

A stone was dropped off a cliff and hit the ground with a speed of 120 ft/s. You may assume that the acceleration due to gravity is −32 ft/s2. Use antiderivatives to find a formula for the velocity at time t. Hint: Since the stone is dropped, what is the initial velocity?

Answer 1

To find the velocity of a stone dropped from a cliff as a function of time, we integrate the acceleration due to gravity. Starting at rest, the velocity function is v(t) = -32t. This represents the velocity of the stone at time t while in free fall, assuming no air resistance.

Step-by-step explanation:

The student is tasked with finding the velocity at time t for a stone dropped off a cliff, given that the acceleration due to gravity is -32 ft/s2 and the stone hits the ground with a speed of 120 ft/s. Since the stone is dropped, the initial velocity is 0 ft/s. To find the velocity as a function of time t, we can use an antiderivative (integral) of the acceleration function.

To solve this, we integrate the acceleration due to gravity to find the velocity function:

1. 
   
3. Let a(t) = -32 ft/s2 be the acceleration due to gravity.
4. 
   
6. The velocity function v(t) is the antiderivative of a(t), so v(t) = ∫ a(t) dt.
7. 
   
9. Integrating -32 with respect to t, we get v(t) = -32t + C, where C is the constant of integration.
10. 
    
12. Since the initial velocity (v(0)) is 0, that means C = 0.
13. 
    
15. Therefore, the velocity function is v(t) = -32t.

This is the formula for the velocity at time t for a stone dropped from rest in free fall, with gravity being the only force acting on it.