You are driving at the speed of 34.3 m/s
(76.7433 mph) when suddenly the car in
front of you (previously traveling at the same
speed) brakes. Considering an average human
reaction, you press the brakes 0.429 s later.
Assume that the brakes on both cars are fully
engaged and that the coefficient of friction is
0.887 between both cars and the road.
The acceleration of gravity is 9.8 m/s
2
.
a) Calculate the magnitude of the acceleration
of the car in front of you when it brakes.
b)Calculate the braking distance for the car in
front of you.
c) Find the minimum safe distance at which you
can follow the car in front of you and avoid
hitting it (in the case of emergency braking
described here).
To solve this problem, we can use the equations of motion and apply the principles of acceleration and friction.
a) To calculate the magnitude of the acceleration of the car in front when it brakes, we can use the equation:
v = u + at,
where:
v = final velocity (0 m/s, as the car comes to a stop),
u = initial velocity (34.3 m/s),
t = time taken to decelerate (which we need to find), and
a = acceleration of the car in front.
We can rearrange the equation as:
t = (v - u) / a.
Since the car in front comes to a stop, v = 0. Plugging in the given values, we have:
t = (0 - 34.3) / a.
Given the value of t (0.429 s), we can solve the equation for a:
a = (0 - 34.3) / t.
a = -34.3 / 0.429,
a ≈ -79.95 m/s².
Therefore, the magnitude of the acceleration of the car in front when it brakes is approximately 79.95 m/s².
b) To calculate the braking distance for the car in front, we can use the equation:
v² = u² + 2as,
where:
v = final velocity (0 m/s),
u = initial velocity (34.3 m/s),
a = acceleration of the car in front (which we just calculated as -79.95 m/s²), and
s = braking distance (which we need to find).
Plugging in the values, we have:
0² = 34.3² + 2(-79.95)s.
Simplifying the equation, we get:
0 = 1174.49 - 159.9s.
Rearranging the equation, we find:
159.9s = 1174.49.
Dividing both sides by 159.9, we have:
s ≈ 7.34 m.
Therefore, the braking distance for the car in front is approximately 7.34 meters.
c) To find the minimum safe distance at which you can follow the car in front and avoid hitting it during emergency braking, you need to account for two factors:
1. Your reaction time (time taken to react after the car in front brakes): 0.429 s.
2. The braking distance required for your car to come to a stop.
The total minimum safe distance can be calculated by adding the braking distance of the car in front (7.34 m) and the braking distance required for your car to stop.
To calculate the braking distance required for your car to stop, we can use:
v² = u² + 2as,
where:
v = final velocity (0 m/s),
u = initial velocity (34.3 m/s),
a = deceleration of your car (which we need to find), and
s = distance traveled by your car to come to a stop.
Rearranging the equation, we have:
a = (0 - 34.3²) / (2s).
Substituting the given values and solving for a, we have:
a = -34.3² / (2s).
Now, we need to find the distance (s) traveled by your car within the reaction time of 0.429 s.
Using the equation:
s = ut + 0.5at²,
where:
u = initial velocity (34.3 m/s),
t = time taken (0.429 s), and
a = deceleration of your car (which we need to find),
we can calculate s:
s = (34.3)(0.429) + 0.5a(0.429)².
Simplifying the equation, we have:
s = 14.718 + 0.0932a.
Substituting this value of s in the previous equation for a, we get:
a = -34.3² / [2(14.718 + 0.0932a)].
Solving this equation will give us the deceleration (a) of your car. Once we have the value of a, we can calculate the braking distance (s) required for your car to stop using the equation:
s = ut + 0.5at².
Finally, the minimum safe distance can be found by adding the braking distance of the car in front (7.34 m) and the braking distance required for your car to stop.