You are driving to the grocery store at 12.7 m/s. You are 108.0 m from an intersection when the traffic light turns red. Assume that your reaction time is 0.470 s and that your car brakes with constant acceleration. How far are you from the intersection when you begin to apply the brakes?

Well, I suppose I am to assume that you stopped in 108 meters.

first you went 12.7*.47 = 5.97 m before your foot hit the brake.

So the brakes must stop you in (108 -6) = 102 meters

That is the answer you asked for.
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The rest of the problem:
102 = 12.7 t - (1/2) a t^2
and
0 = 12.7 - a t
t = 12.7/a
so
102 = 12.7^2/a - (1/2) (12.7^2)/a
204 a = 12.7^2
a = .8 m/s^2 about a tenth of g

To find the distance from the intersection when you begin to apply the brakes, we can break down the problem into two parts: the reaction time and the braking distance.

1. Reaction time:
Your reaction time is given as 0.470 s. This means that it takes you 0.470 seconds to notice the change in traffic light and start applying the brakes.

During this time, your car will still be traveling at a constant speed. To find the distance covered during the reaction time, we use the formula:

Distance = Speed × Time

Given that your speed is 12.7 m/s and the reaction time is 0.470 s:

Distance (reaction time) = 12.7 m/s × 0.470 s

2. Braking distance:
Once you start applying the brakes, your car will decelerate with constant acceleration until it comes to a complete stop. We need to find the distance covered during this deceleration.

To find the braking distance, we can use the kinematic equation:

Distance = (Initial Velocity × Time) + (0.5 × Acceleration × Time^2)

In this case, the initial velocity is 12.7 m/s, the final velocity is 0 (as the car comes to a complete stop), and the time is the difference between the reaction time and the total time it takes to stop.

To calculate the braking time, we need to find the total time it takes for the car to come to a stop. Assuming that the acceleration during braking is constant, we have:

Final Velocity = Initial Velocity + (Acceleration × Time)

Given that the final velocity is 0 m/s, the initial velocity is 12.7 m/s, and the acceleration is unknown:

0 = 12.7 m/s + (Acceleration × Time to stop)

Solving for Time to stop:

Time to stop = -12.7 m/s / Acceleration

Now, we can substitute Time to stop back into the braking distance equation:

Distance (braking) = (12.7 m/s × Time to stop) + (0.5 × Acceleration × (Time to stop)^2)

3. Total distance:
Finally, to find the total distance from the intersection when you begin to apply the brakes, we add the distance traveled during the reaction time to the braking distance:

Total distance = Distance (reaction time) + Distance (braking)