Q:

My friend recently got a speeding ticket that is questionable. Here is why. The policeman told him that he clocked him going 83 mph. My friend was at a rest station, starting at zero velocity, pulled onto the highway and travelled for 1 mile where he had to come to a complete stop to turn off the highway. I was wondering how long it would take him to reach 83 mph or at what distance would that occur and how far from the one mile point would he have to begin decelerating to be at a complete stop at the one mile point. He is hoping that this will help him in his defense. Do we have enough information to figure this out?
I would appreciate any help you could provide.

- Ashley (age 16)

- Ashley (age 16)

A:

Ashley -

Without knowing the rate at which your friend's car accelerates or deccelerates, there's no way to figure out exactly how far the car would need to go to get up to 83 mph or back to 0. But we can answer this question... Can the car get up to 83 and back to 0 within 1 mile?

First, we'll assume the car spends exactly 1/2 mile accelerating and 1/2 mile deccelerating. So the second 1/2 mile is exactly the same as the first 1/2 mile, just with the car slowing down instead of speeding up. We'll also assume that the car accelerates and deccelerates at a constant rate. In this case we can use a couple of equations about how things move:

total distance = 1/2 x acceleration x time x time

final velocity = acceleration x time

Plugging in your numbers for just the first 1/2 mile (accelerating up to 83), we have:

1/2 = 1/2 x acceleration x time x time

83 = acceleration x time

We can plug the second equation into the first one to get:

time = 1/83 hrs (43 seconds)

So your friend would have to be able to go from 0 to 83 in 43 seconds. For the second 1/2 mile, it's the opposite - he has to go from 83 to 0 in 43 seconds. Sine this is a fairly reasonable idea, I think you should start looking for another defense.

-Tamara & Tim

Without knowing the rate at which your friend's car accelerates or deccelerates, there's no way to figure out exactly how far the car would need to go to get up to 83 mph or back to 0. But we can answer this question... Can the car get up to 83 and back to 0 within 1 mile?

First, we'll assume the car spends exactly 1/2 mile accelerating and 1/2 mile deccelerating. So the second 1/2 mile is exactly the same as the first 1/2 mile, just with the car slowing down instead of speeding up. We'll also assume that the car accelerates and deccelerates at a constant rate. In this case we can use a couple of equations about how things move:

total distance = 1/2 x acceleration x time x time

final velocity = acceleration x time

Plugging in your numbers for just the first 1/2 mile (accelerating up to 83), we have:

1/2 = 1/2 x acceleration x time x time

83 = acceleration x time

We can plug the second equation into the first one to get:

time = 1/83 hrs (43 seconds)

So your friend would have to be able to go from 0 to 83 in 43 seconds. For the second 1/2 mile, it's the opposite - he has to go from 83 to 0 in 43 seconds. Sine this is a fairly reasonable idea, I think you should start looking for another defense.

-Tamara & Tim

*(published on 10/22/2007)*