Determine the stopping distances for a car with an initial speed of 93 km/h & human reaction time of 0.80 s, for an acceleration a=−4.0m/s2.?
Vi = 93000 m/3600 s
d = d coast + d braking
d coast = Vi (.8)
v = Vi + a t
0 = Vi - 4 t
solve for t
then
d braking = Vi t -(1/2)(4) t^2
To determine the stopping distance for a car with an initial speed and acceleration, we need to consider two main factors: the distance covered during the driver's reaction time and the distance covered during the braking phase.
1. Distance covered during the reaction time:
The reaction time is the time it takes for the driver to recognize a hazard and react by applying the brakes. In this case, the reaction time is given as 0.80 seconds. During this time, the car continues to move forward at its initial speed. To calculate the distance covered during the reaction time, we use the formula:
distance = speed × time
distance = (93 km/h) × (0.80 s)
However, it's important to note that the initial speed is given in kilometers per hour (km/h) while the time is given in seconds (s). To convert the speed from km/h to m/s, we need to divide it by 3.6:
speed = (93 km/h) / (3.6)
speed = (25.83 m/s)
Therefore, the distance covered during the reaction time is:
distance = (25.83 m/s) × (0.80 s)
distance = (20.664 m)
2. Distance covered during the braking phase:
Once the driver applies the brakes, the car starts decelerating due to the given acceleration of -4.0 m/s^2. We can use the following formula to calculate the distance covered during the braking phase:
distance = (speed^2) / (2 * acceleration)
Again, we need to convert the speed from km/h to m/s:
speed = (93 km/h) / (3.6)
speed = (25.83 m/s)
Now we can calculate the distance:
distance = (25.83 m/s)^2 / (2 * (-4.0 m/s^2))
distance = (25.83^2 m^2/s^2) / (-8.0 m/s^2)
distance = (667.2489 m^2/s^2) / (-8.0 m/s^2)
distance = -83.4061 m
Note that the distance is negative because the car is decelerating.
3. Total stopping distance:
To calculate the total stopping distance, we add the distance covered during the reaction time to the distance covered during the braking phase:
total stopping distance = distance during reaction time + distance during braking phase
total stopping distance = 20.664 m + (-83.4061 m)
total stopping distance = -62.7411 m
Therefore, the stopping distance for the car with an initial speed of 93 km/h, a human reaction time of 0.80 s, and an acceleration of -4.0 m/s^2 is approximately -62.7411 meters. The negative sign indicates that the car will come to a stop in the opposite direction of its initial motion.