You are driving to the grocery store at 18.2 m/s. You are 120.0 m from an intersection when the traffic light turns red. Assume that your reaction time is 0.270 s and that your car brakes with constant acceleration.How far are you from the intersection when you begin to apply the brakes?
To determine how far you are from the intersection when you begin to apply the brakes, we need to consider the time it takes for you to react to the red light and the distance traveled during that time.
Given information:
Initial velocity (v0) = 18.2 m/s
Distance to intersection (s) = 120.0 m
Reaction time (t_r) = 0.270 s
First, let's calculate the distance traveled during the reaction time:
Distance traveled during reaction time (d_r) = v0 * t_r
d_r = 18.2 m/s * 0.270 s
d_r = 4.914 m
Now, let's calculate the effective initial distance by subtracting the distance traveled during the reaction time from the total distance to the intersection:
Effective initial distance (s') = s - d_r
s' = 120.0 m - 4.914 m
s' = 115.086 m
Now, we can use the formula for distance traveled with constant acceleration:
Distance traveled (s_t) = v0 * t + (1/2) * a * t^2
In this case, the initial velocity is v0 = 0 m/s (as you've begun braking), and we need to calculate the acceleration (a). We know that the distance traveled is (s' = 115.086 m), and the time taken to come to rest is the reaction time (t_r = 0.270 s). We can rearrange the formula to solve for the acceleration:
a = (2 * (s' - v0 * t_r)) / t_r^2
a = (2 * (115.086 m - 0 m/s * 0.270 s)) / (0.270 s)^2
a = (2 * 115.086 m) / 0.0729 s^2
a = 3165.1 m/s^2 (rounded to four decimal places)
Now we have the acceleration value, and we can use it to calculate the distance travelled from the point of braking until the car comes to a stop:
Distance traveled during braking (d_b) = (1/2) * a * t_r^2
d_b = (1/2) * 3165.1 m/s^2 * (0.270 s)^2
d_b = 113.19 m (rounded to two decimal places)
Finally, to determine the distance from the intersection when you begin to apply the brakes, we add the distance traveled during the reaction time (d_r) to the distance traveled during braking (d_b):
Distance from intersection when brakes are applied = s' + d_b
Distance from intersection when brakes are applied = 115.086 m + 113.19 m
Distance from intersection when brakes are applied = 228.276 m (rounded to three decimal places)
Therefore, you are approximately 228.276 m from the intersection when you begin to apply the brakes.