If you are travelling in your Bugatti at 45 m/s (about 100 mi/hr) and step on the brakes, creating an acceleration of - 12.5 m/s^2. How far do you travel before stopping? How far would it be if you were only travelling 22.5 m/s (half as fast) ? Discuss the change in distance that occurs when you halve the speed.

You travel 81 m before you stop.

If you were traveling at 22.5 m/s you will travel 20.25 m until you stop.

When you travel half as fast, the traveled distance until you come to stop is four times less (81 m / 4 = 20.25 m).

This is so because the position is a quadratic function with respect to time. If you halve the time needed to come to stop (by halving the velocity), the distance will be divided by 2² = 4. If the velocity would have been reduced by 3, then the distance would be reduced by 3² = 9 (check it out!). In contrast, the velocity is a linear function, that's why when you halve the speed, the time it takes for you to stop is halved too.

Explanation:

The equations for the position and velocity of an object moving along a straight line are as follows:

x = x0 + v0 · t + 1/2 · a · t²

v = v0 + a · t

Where:

x = position at time t

x0 = initial positon

v0 = initial velocity

t = time

a = acceleration

v = velocity at time t

First let's calculate how much time it takes for you to stop. For this, we are going to use the equation for velocity knowing that when you stop, your velocity is 0:

v = v0 + a · t

0 = 45 m/s - 12.5 m/s² · t

-45 m/s / - 12.5 m/s² = t

t = 3.6 s

Now, we can calculate how much distance you travel in that time:

(let's consider the origin of the frame of reference your position when you apply the brakes)

x = x0 + v0 · t + 1/2 · a · t²

x = 0 m + 45 m/s · 3.6 s - 1/2 · 12.5 m/s² · (3.6 s) ²

x = 81 m

Let's do the same calculations but with v0 = 22.5 m/s:

v = v0 + a · t

0 = 22.5 m/s - 12.5 m/s² · t

-22.5 m/s / - 12.5 m/s² = t

t = 1.8 s

x = x0 + v0 · t + 1/2 · a · t²

x = 0 m + 22.5 m/s · 1.8 s - 1/2 · 12.5 m/s² · (1.8 s) ²

x = 20.25 m

When you travel half as fast, the traveled distance until you come to stop is four times less (81 m / 4 = 20.25 m).

This is so because the position is a quadratic function with respect to time. If you halve the time needed to come to stop (by halving the velocity), the distance will be divided by 2² = 4. If the velocity would have been reduced by 3, then the distance would be reduced by 3² = 9 (check it out!). In contrast, the velocity is a linear function, that's why when you halve the speed, the time it takes for you to stop is halved too.