A driver traveling at 50 mph is 80 from a wall
ahead.
If
the
driver
applies
the
brakes
immediately and begins slowing the vehicle at
10 m/sec2 (decelerating), find the PIEV time
when the distance of the car from the wall when
it stop is 10.28 m.
To find the PIEV time (time it takes for the car to come to a stop) given the initial distance and deceleration, we can use the following steps:
1. Convert the units of the given values to a consistent unit. In this case, since the deceleration is given in m/sec^2, let's convert the initial distance from miles to meters and the speed from mph to m/s for easier calculations:
- Initial distance: 80 miles = 80 * 1609.34 meters = 128747.2 meters (rounded to the nearest tenth)
- Speed: 50 mph = 50 * 0.44704 meters/second = 22.352 meters/second (rounded to the nearest thousandth)
2. Determine the time it takes for the car to stop. We know that the final distance is 10.28 meters, and the initial distance is 128747.2 meters. The deceleration is given as 10 m/sec^2.
- We can use the SUVAT equation to calculate the time (t) it takes for the car to stop:
v^2 = u^2 + 2as
Where:
v = final velocity (which is 0 m/s since the car comes to a stop)
u = initial velocity (22.352 m/s)
a = acceleration (deceleration of -10 m/s^2)
s = distance (10.28 meters)
Rearranging the equation, we get:
2as = -u^2
t = (-u ± √(u^2 - 4as)) / (2a)
Plugging in the values:
t = (-22.352 ± √(22.352^2 - 4 * -10 * 10.28)) / (2 * -10)
t = (-22.352 ± √(499.370304 - 411.2)) / (-20)
t = (-22.352 ± √(88.170304)) / (-20)
t = (-22.352 ± 9.385971) / (-20)
The positive value of t will give us the time it takes for the car to stop, so:
t = (-22.352 + 9.385971) / (-20)
t = -12.96603 / (-20)
t = 0.648 seconds (rounded to the nearest thousandth)
3. Therefore, the PIEV time when the car stops at a distance of 10.28 meters is approximately 0.648 seconds.