The driver of a 2.0 × 103 kg red car traveling on the highway at 45m/s slams on his brakes to avoid striking a second yellow car in front of him, which had come to rest because of blocking ahead as shown in above Fig. After the brakes are applied, a constant friction force of 7.5 × 103 N acts on the car. Ignore air resistance.
(a) Determine the least distance should the brakes be applied to avoid a collision with the other vehicle?
(b) If the distance between the vehicles is initially only 40.0 m, at what speed would the collision occur
To solve this problem, we can use the equations of motion and the concept of work done by friction. Here's how you can find the answers to parts (a) and (b):
(a) To determine the least distance required to avoid a collision, we need to calculate the deceleration of the car using the given information.
We know the mass of the car (m = 2.0 × 10^3 kg), the initial velocity (v = 45 m/s), and the friction force (F = 7.5 × 10^3 N). We can use Newton's second law of motion, which states that the net force acting on an object is equal to its mass times its acceleration (Fnet = ma).
Considering the direction of the motion, the net force acting on the car is the difference between the friction force and the force required to stop the car (which is opposite in direction to the motion). So, we have:
Fnet = Ffriction - Fstop
Using the equation Fnet = ma, we find:
ma = Ffriction - Fstop
Solving for acceleration (a):
a = (Ffriction - Fstop) / m
Substituting the values given:
a = (7.5 × 10^3 N - Fstop) / (2.0 × 10^3 kg)
Now, to find the minimum distance required to stop the car, we can use the equation of motion:
v^2 = u^2 + 2as
where v is the final velocity (which is 0 in this case because the car comes to a stop), u is the initial velocity (45 m/s), a is the acceleration, and s is the distance traveled.
Rearranging the equation, we get:
s = (v^2 - u^2) / (2a)
Substituting the values:
s = (0 - (45 m/s)^2) / (2 * a)
Calculate the acceleration (a) using the earlier equation and substitute it into this equation to find the minimum distance (s) required for the car to stop.
(b) To find the speed at which the collision would occur if the initial distance between the cars is 40.0 m, we need to determine the time it takes for the red car to cover this distance.
Since the initial velocity of the yellow car is 0 (it's at rest), the time taken by the red car to cover a distance of 40.0 m can be found using the equation of motion:
s = ut + (1/2)at^2
In this case, s is 40.0 m, u is the initial velocity (45 m/s), and a is the acceleration (calculated earlier). Solve this equation for time (t).
Then, to find the speed at which the collision would occur when the distance becomes zero (both cars are in the same position), apply the equation of motion:
v = u + at
where v is the final velocity (which is what we need to find), u is the initial velocity, a is the acceleration, and t is the time calculated earlier. Substitute the values and solve for v to get the speed at which the collision would occur.
By following these steps, you can find the least distance required to avoid a collision and the speed at which the collision would occur.