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MathCalculus

What Is the Mean Value Theorem? Apply It to a Falling Rock

The rock hits after 5/2 s, its average velocity is −40 ft/s, and the Mean Value Theorem identifies c = 5/4 s when instantaneous velocity matches it.

Question

A rock is dropped from a height of 100 ft. Its position tt seconds after release, until it hits the ground, is

s(t)=−16t2+100.s(t)=-16t^2+100.
  1. Determine how long it takes the rock to hit the ground.
  2. Find the average velocity from release until impact.
  3. Find the time guaranteed by the Mean Value Theorem when the instantaneous velocity equals that average velocity.

Answer

1. Find the impact time

The rock reaches the ground when s(t)=0s(t)=0:

−16t2+100=0⟹t2=254.-16t^2+100=0 \quad\Longrightarrow\quad t^2=\frac{25}{4}.

Time is nonnegative, so

t=52 s.\boxed{t=\frac52\text{ s}}.

2. Find the average velocity

Over [0,5/2][0,5/2],

vavg=s(5/2)−s(0)5/2−0=0−1005/2=−40 ft/s.v_{\text{avg}} =\frac{s(5/2)-s(0)}{5/2-0} =\frac{0-100}{5/2} =\boxed{-40\text{ ft/s}}.

3. Apply the Mean Value Theorem

The polynomial s(t)s(t) is continuous on [0,5/2][0,5/2] and differentiable on (0,5/2)(0,5/2), so the theorem guarantees a cc in that open interval with s′(c)=vavgs'(c)=v_{\text{avg}}.

Since

s′(t)=−32t,s'(t)=-32t,

solve

−32c=−40⟹c=54 s.-32c=-40 \quad\Longrightarrow\quad c=\boxed{\frac54\text{ s}}.

At 1.251.25 seconds after release, the instantaneous velocity is −40-40 ft/s, equal to the average velocity over the full fall.

Evidence boundary

This answer follows the stated position model from release through impact and treats velocity as signed vertical velocity. The model does not include air resistance; the Mean Value Theorem conclusion is conditional on the given differentiable model.

Sources

These references support the concepts and methods used in the explanation above.

What Is the Mean Value Theorem? Falling Rock Answer | Verla