Answer:
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.
Step-by-step 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.