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.