The speed limit in a school zone is 40 km/hr (11.1 m/s). A driver traveling at this speed sees a child run into the road 13 m ahead of his car. He applies the brakes at a uniform rate of -8.0 m/s2. The driver has a reaction time of 0.25 sec prior to stepping on the brake pedal.
a. Determine the distance the driver travels during their reaction time
b. Will the driver hit the child? Yes or no. Show calculations to support your answer.
Vi = 11.1 m/s
v = 11.1 - 8 t
d = .25 *11.1 = 2.775 m during reaction time
now the brakes are on
v = 0 at what additional time?
8 t = 11.1
t = 1.39 seconds more
x during braking = average speed during stop * time
= 11.1 / 2 * 1.39 = 7.70 meters more
so stop distance = 2.775 + 7.70 = 10.47 meters total to stop
Whew, no problem :)
a. To determine the distance the driver travels during their reaction time, we can use the formula:
Distance = Initial velocity * Time + 0.5 * Acceleration * Time^2
Given:
Initial velocity (v0) = 11.1 m/s
Time (t) = 0.25 sec (reaction time)
Acceleration (a) = 0 m/s^2 (since the car is not yet braking during the reaction time)
Plugging in the values, we get:
Distance = 11.1 m/s * 0.25 s + 0.5 * 0 m/s^2 * (0.25 s)^2
Distance = 2.775 m + 0 m
Distance = 2.775 m
Therefore, the driver travels a distance of 2.775 meters during their reaction time.
b. To determine if the driver will hit the child, we need to calculate the stopping distance of the car from the point when the brakes are applied.
First, let's determine the time taken for the car to come to a complete stop (final velocity = 0 m/s) using the following formula:
Final velocity (vf) = Initial velocity (v0) + (Acceleration (a) * Time taken to stop (t_stop))
Since the final velocity is 0 m/s and the initial velocity is 11.1 m/s, we have:
0 m/s = 11.1 m/s + (-8.0 m/s^2) * t_stop
Simplifying this equation, we find:
8.0 m/s^2 * t_stop = 11.1 m/s
t_stop = 11.1 m/s / 8.0 m/s^2
t_stop = 1.3875 s
Now, let's calculate the stopping distance using the formula:
Stopping distance = Initial velocity * Time taken to stop + 0.5 * Acceleration * (Time taken to stop)^2
Plugging in the values, we get:
Stopping distance = 11.1 m/s * 1.3875 s + 0.5 * (-8.0 m/s^2) * (1.3875 s)^2
Stopping distance = 15.35375 m - 7.241 m
Stopping distance = 8.11275 m
The stopping distance is 8.11275 meters.
Since the child is 13 meters ahead and the driver's reaction time distance is 2.775 meters, the total distance covered by the driver before coming to a stop is:
Total distance = Reaction time distance + Stopping distance
Total distance = 2.775 m + 8.11275 m
Total distance = 10.88775 m
Comparing the total distance covered by the driver to the distance of the child, we can conclude:
If the total distance is greater than or equal to the distance of the child, then the driver will stop in time and not hit the child. Otherwise, if the total distance is less than the distance of the child, then the driver will hit the child.
In this case, the total distance (10.88775 m) is less than the distance of the child (13 m), so the driver will hit the child.