During a move, Driver A and Driver B drove two separate cars from their home town, Washington D.C. to New York City. It took them about 5 hours to travel to NYC. They agreed to travel no faster than 80mph during the trip. The total distance between the two places is 240 miles. Using the Intermediate Value Theorem, show that they must drive at the same speed at one time or another during the entire journey.
I am not completely sure what I should be doing for this question. Typically I see a given function within the questions, and a request to find the root or a given point within the function. I do know that velocity is a continuous function, so IVT applies. I also know that regardless of the speeds traveled, the speed of the slower of the two cars will be achieved by the faster car at least at the tail-end of the trip during deceleration. I am confused as to how to portray this information, if it is even the correct information to portray. Thank you in advance.
Best Answer
I think that the relative starting times does not matter.
Let car A start at $t=t_1$ and B start at $t=t_2$. We assume that both cars take 5h to travel $240$ miles. Just for the sake of it, average speed is 48 mph. Let $v_1(\Delta t)$ and $v_2(\Delta t)$ be instantaneous speeds of A and B at $t=t_1+\Delta t$ and $t=t_2+\Delta t$ respectively, where $0\leq \Delta t\leq 5h$. Since in a real-world scenario, speed must be continuous, hence $v_1,v_2$ and thus $v_1-v_2$ are continuous functions.
Consider $v_1(\Delta t)\gt v_2(\Delta t)$ for all $\Delta t.$ This implies that $$\int_0^{5h}v_1d(\Delta t)>\int_0^{5h}v_2d(\Delta t)$$ which implies that the distance travelled by A in 5 hours is greater than that by B in 5 hours, which is clearly not the case as they are equal. You can similarly do the vice-versa.
This implies that there must exist at least one value of $\Delta t$ such that $v_1(\Delta t)>v_2(\Delta t)$ AS WELL AS at least one value of $\Delta t$ such that $v_1(\Delta t)\lt v_2(\Delta t)$.
Thus, the function $v_1-v_2>0$ for some $\Delta t$ as well as $v_1-v_2<0$ for some other $\Delta t$ on the interval $\Delta t\in [0,5hrs]$. Thus, by the intermediate value theorem, $v_1-v_2$ must have a zero in $\Delta t\in[0,5h]$. Hence proved.
The point is not that they will have the same velocity at one instant of time. The point is that they will have the same velocity corresponding to the same time interval $\Delta t$ after they start.
This will change to having same speeds at same instants of time if they start together.