mean value theorem problem with no equation. Only given average velocity.

Question

I have a question that states a person traveled continuously for $4$ hours and a total of $224$ miles, making the average speed $56$ mph. Prove that the car was traveling exactly the average speed of $56$ mph in at least one instant.

How can I prove this?

Answer 1

Let $s(t)$ the distance at the time $t$, $t\in [0\text{hr}, 4\text{hr}]$. From physics, we know that the velocity is given by: $v(t)=s'(t)$. Also, $s(0\text{hr})=0\text{mi}$ and $s(4\text{hr})=224\text{mi}$.

Now, apply the mean value theorem to the interval above: we must have $t_0\in [0\text{hr}, 4\text{hr}]$ such that:

$$v(t_0)=s'(t_0)=\frac{s(4\text{hr})-s(0\text{hr})}{4\text{hr}-0\text{hr}}=\frac{224\text{mi}-0\text{mi}}{4\text{hr}-0\text{hr}}=\frac{224\text{mi}}{4\text{hr}}=56\text{mph}$$

so the velocity (and speed) at $t_0$ was $56\text{mph}$, as claimed.

Answer 2

First, suppose the car starts with a speed less than $56$ mph. Then, suppose for a contradiction that there is no instant such that average speed of the car is $56$ mph. Notice that it travels continuously, so this is not possible because it has an average speed of $56$ mph and you cannot have that average with values all of which are less than $56$. So we have a contradiction as required and car has $56$ mph speed at some instant.

Can you do the same for the case where car starts with a speed larger than $56$ mph?

Answer 3

This answer requires that you can prove the following.

If $f$ is continuous on $[a, b]$ and $f(a) < 0 < f(b)$, then there is some number $x$ in $[a, b]$ with $f(x) = 0$.

The car must have traveled either faster or slower than the average speed during its trip, or it would have only traveled at the average speed. Because the speed of the car is continuous, it must have taken on every value between its bounds, including the average speed.

Answer 4

For fun:

Assume the speed is a continuos function of time, I.e. from 58mph it cannot jump to 54 mph (say) abruptly .

Assume the statement is false.

Means : There is no point in time such that the speed was 56 mph.

Then it was greater or less than 56 mph all the time.

(You need continuity here) .

1) If it was always faster than 56 mph you get there faster.

2) Similar argument for slower than 56 mph, more time .