The speed limit in a school zone is 40Km/h. A driver traveling at this speed sees a child run into the road 13m ahead of his car. He applies the breaks, and the car decelerates at a uniform rate of 0.8m/s². If the driver's reaction time is 0.25 seconds, what distance will the car travel before coming to rest?
To find the distance the car will travel before coming to rest, we need to calculate two distances: the distance traveled during the driver's reaction time and the distance traveled while decelerating.
1. Distance during the driver's reaction time:
During the driver's reaction time, the car continues to travel at its initial speed without any deceleration. The formula to calculate this distance is:
Distance = Initial speed x Time
Given that the initial speed is 40 km/h, we need to convert it to meters per second:
40 km/h = (40 x 1000) m / (60 x 60) s = 11.11 m/s
The reaction time is given as 0.25 seconds, so the distance during the reaction time is:
Distance_reaction = 11.11 m/s x 0.25 s = 2.78 m
2. Distance while decelerating:
To calculate the distance traveled while decelerating, we need to use the formula of motion:
Distance = (Initial velocity x Time) + (0.5 x Acceleration x Time²)
The initial velocity is 11.11 m/s (as calculated before).
The acceleration is -0.8 m/s² because the car is decelerating (negative sign indicates deceleration).
We need to find the time taken to stop.
Using the formula of motion, the equation becomes:
0 = 11.11 m/s x t + 0.5 x (-0.8 m/s²) x t²
Rearranging the equation to solve for time:
-0.8t² + 11.11t = 0
t(-0.8t + 11.11) = 0
Since time (t) cannot be zero, the equation -0.8t + 11.11 must be zero to solve for time:
-0.8t + 11.11 = 0
-0.8t = -11.11
t = -11.11 / -0.8
t = 13.89 seconds
Now, we can calculate the distance while decelerating:
Distance_deceleration = (11.11 m/s x 13.89 s) + (0.5 x (-0.8 m/s²) x (13.89 s)²
Simplifying the equation:
Distance_deceleration = (154.20 m) + (-7.85 m)
Distance_deceleration = 146.35 m
3. Total distance traveled:
The total distance traveled is the sum of the distance during the reaction time and the distance while decelerating:
Total distance = Distance_reaction + Distance_deceleration
Total distance = 2.78 m + 146.35 m
Total distance = 149.13 m
Therefore, the car will travel a distance of 149.13 meters before coming to rest.