# Car Acceleration: Top Speed, Travel Distance & Mile-Long Race

• DustyGeneral
In summary: You know that the car starts from rest, so the initial velocity is 0. Use that to solve for the constant.And you're welcome, glad I could help!In summary, the problem presents a car accelerating from rest with a speed given by v(t)=vm(1-e-at). Part a asks for the top speed of the car, which is when t=∞ and the maximum value of v(t) is equal to vm. Part b inquires about the distance traveled in time t, which is given by x(t)=vm*((e-at)/a+t), and the constant of integration can be found by setting the initial velocity to 0. Part c sets up a race between two cars, one starting
DustyGeneral
Hello all, here is my problem:

A car accelerates, starting from rest, with a speed that is given by:

v(t)=vm(1-e-at)

a) What is the top speed of the car? Explain why.

b) How far does the car travel in time t?

c) Suppose the car can accelerate from 0 to 60 mph in 2.9s, and has a top speed of 195 mph. Imagine a mile-long race between two of these cars, with the same finish line, but with a different starting line: One drives along the ground towards the starting line at a point one-mile away, while the other is dropped out of a plane one mile above the ground. Which one reaches the finish line first? (ignore any air resistance for the falling car).

That is the entire problem. I just need help getting started.

First, what does the vm represent in the speed equation?
Second, how exactly do I calculate the top speed?

I also suppose that I will need the position and acceleration equations which I derived from speed equation for b and c respectively.

I got:

x(t)=vm*((e-at)/a+t)
v(t)=vm*(1-e-at)
a(t)=vm*(ae-at)

DustyGeneral said:
First, what does the vm represent in the speed equation?
Think of it as a constant. Once you answer part a, the notation will make more sense.

Second, how exactly do I calculate the top speed?
You won't be able to get a numerical value, if that's what's throwing you off. Answer in terms of the constants given.

You are given the speed as a function of time. If you graphed it, what would that function look like? What's its maximum value?

I know I won't get a numerical value as the top speed. I just do not know what path to take to manipulate v(t) to get top speed.

DustyGeneral said:
I just do not know what path to take to manipulate v(t) to get top speed.
Does the speed increase or decrease as time increases?

What's the maximum value of 1-e-at ?

Increase in time=Increase in speed.

Max value of 1-e-at = 0

DustyGeneral said:
Increase in time=Increase in speed.
Good!

Max value of 1-e-at = 0
Really? So the car doesn't even move? (Think that one over. )

DustyGeneral said:
Increase in time=Increase in speed.

Max value of 1-e-at = 0

Are you sure?

Nevermind. Duh. I was thinking something completely different. So what is the a in the exponent? Acceleration due to gravity? If so then the max value would be 1, but that doesn't make sense.

DustyGeneral said:
So what is the a in the exponent? Acceleration due to gravity?
Treat 'a' as just another constant.

If so then the max value would be 1, but that doesn't make sense.
Makes sense to me. At what time will the car have max speed? What is that max speed?

So what I'm getting is that:

v(t)=vm(1-e-at)

We've said that the max value for (1-e-at) = 1

So that means at max v(t)=vm

In order to get that t=∞.

DustyGeneral said:
So what I'm getting is that:

v(t)=vm(1-e-at)

We've said that the max value for (1-e-at) = 1

So that means at max v(t)=vm

In order to get that t=∞.
Perfect! As time goes on, the speed gets closer to the maximum value of vm.

Perhaps now you can guess what the 'm' stands for.

I figured that's what it stood for I seem to have a habit of jumping to conclusions like that early on in the problem and confuse my self because I get determined to make that scenario be the case (right or wrong.)

So part a is understood. Part b would be the distance equation setting t=∞?

DustyGeneral said:
Part b would be the distance equation setting t=∞?
They don't want the distance at infinite time, but at time "t". (You've already solved that one in your first post.)

1 person
See what I mean? Get ahead of myself and flustered and try to make a certain scenario fit. Then for part c I am given acceleration, the time for that acceleration, and top speed. Plug and chug then. Very helpful Al.

Doc Al said:
(You've already solved that one in your first post.)
Correction: You've almost solved it. Don't forget about the constant of integration.

## What is car acceleration?

Car acceleration is the rate at which a car's velocity changes over time. It is typically measured in miles per hour per second (mph/s) or meters per second squared (m/s²).

## How is top speed determined?

The top speed of a car is determined by its engine power, aerodynamics, and weight. The more powerful the engine and the lighter the car, the higher the top speed it can achieve. Aerodynamics also play a role as air resistance can limit a car's speed.

## What factors affect a car's acceleration?

The main factors that affect a car's acceleration are its engine power, weight, and traction. A more powerful engine and lighter weight can improve acceleration, while factors like road conditions and tire grip can impact traction and ultimately affect acceleration.

## How far can a car travel in one mile at top speed?

The distance a car can travel in one mile at top speed depends on its top speed and the time it takes to reach that speed. For example, a car with a top speed of 100 mph that takes 10 seconds to reach that speed can travel approximately 0.277 miles in one mile at top speed.

## What is a mile-long race and how is it different from other races?

A mile-long race is a race that covers a distance of one mile. It is different from other races, such as a drag race or a track race, in that it is a set distance rather than a set time. This means that the car's acceleration and top speed are the main factors that determine the winner.

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