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.