You are driving at the speed of 33.4 m/s
(74.7296 mph) when suddenly the car in
front of you (previously traveling at the same speed) brakes. Considering an average human reaction, you press your brakes 0.484 s later. Assume that the brakes on both cars are fully
engaged and that the coefficient of friction is 0.92 between both cars and the road. The acceleration of gravity is 9.8 m/s2 .
a) Calculate the braking distance for the car in front of you.
b) 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).
The braking time is Vo/a, where a is the the deceleration rate I have already provided in a previous post.
The braking distance is Vo^2/(2a)
Write equations for position vs time for both cars, with following distance as an variable. Solve for the following distance that just avoids collision.
To solve this problem, we can use the following steps:
Step 1: Calculate the deceleration of the car in front of you.
The deceleration of the car can be calculated using the equation:
a = (v - u) / t
Where:
- a is the deceleration (in m/s^2),
- v is the final velocity of the car (which is 0 m/s in this case),
- u is the initial velocity of the car (which is 33.4 m/s in this case), and
- t is the time taken for the car to decelerate (which is the reaction time plus the time taken to stop).
Given:
- v = 0 m/s,
- u = 33.4 m/s, and
- t = reaction time + time taken to stop.
Step 2: Calculate the reaction time.
The reaction time can be calculated by multiplying the time taken to react (0.484 s in this case) by the initial velocity (33.4 m/s).
reaction time = t_react * u
Given:
- t_react = 0.484 s,
- u = 33.4 m/s.
Step 3: Calculate the time taken to stop.
The time taken to stop can be calculated using the equation:
t_stop = u / a
Given:
- u = 33.4 m/s,
- a (the deceleration) = calculated in Step 1.
Step 4: Calculate the braking distance for the car in front of you.
The braking distance can be calculated using the equation:
s = (u^2) / (2a)
Given:
- u = 33.4 m/s,
- a (the deceleration) = calculated in Step 1.
Step 5: Calculate the minimum safe distance.
The minimum safe distance can be calculated by summing the braking distance for the car in front and the distance traveled during the reaction time of the driver.
minimum safe distance = braking distance + (reaction time * initial velocity)
Given:
- braking distance = calculated in Step 4,
- reaction time = 0.484 s,
- initial velocity = 33.4 m/s.
Using the above steps and calculations, we can now find the answers to the given problem.