The brakes are applied to a car traveling on a dry, level highway. A typical value for the magnitude of the car's acceleration is 5.35 m/s2. If the car's initial speed is 29.1 m/s, how long does it take to stop and how far does it travel, starting from the moment the brakes are applied?
To find out how long it takes for the car to stop and how far it travels, we first need to identify the information given and the equation(s) required to solve the problem.
Given:
- Magnitude of the car's acceleration (a) = 5.35 m/s^2
- Initial speed (u) = 29.1 m/s
- Final speed (v) = 0 m/s (car stops)
We can use the following kinematic equation to solve for the time taken (t) and the distance traveled (s):
v^2 = u^2 + 2as
Substituting the known values:
0^2 = (29.1)^2 + 2(5.35)s
Simplifying the equation:
0 = 843.81 + 10.7s
Rearranging the equation:
10.7s = -843.81
Dividing by 10.7:
s = -78.923 m
Since distance cannot be negative, we ignore the negative sign and take the magnitude of the distance:
s = 78.923 m
This means the car travels 78.923 meters before coming to a stop.
Now, to find the time taken:
v = u + at
Substituting the known values:
0 = 29.1 + 5.35t
Simplifying the equation:
5.35t = -29.1
Dividing by 5.35:
t = -5.428 s
Again, since time cannot be negative, we ignore the negative sign and take the magnitude of the time:
t = 5.428 s
This means it takes approximately 5.428 seconds for the car to come to a stop.
Therefore, the car travels approximately 78.923 meters and takes approximately 5.428 seconds to stop, starting from the moment the brakes are applied.