A drag racer crosses the finish line of a 400.0-m track with a final speed of 104 m/s. (a) Assuming constant acceleration during the race, find the racer’s time and the minimum co efficient of static friction between the tires and the road. (b) If, because of bad tires or wet pavement the acceleration were 30.0% smaller, how long would it take to finish the race?
7.69 sec
To find the answers to these questions, we can use the equations of motion. The main equation that we will be using is the following:
v^2 = u^2 + 2as
where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement.
(a) To find the racer's time and the minimum coefficient of static friction, we need to first find the initial velocity and the acceleration.
We are given the final speed, v = 104 m/s, and the track length, s = 400.0 m.
Using the equation above, we have:
104^2 = u^2 + 2a(400.0)
10816 = u^2 + 800a
We also know that the initial velocity, u, is 0 because the racer starts from rest. Therefore, the equation becomes:
10816 = 0 + 800a
Simplifying, we find:
800a = 10816
a = 13.52 m/s^2
Now that we have the acceleration, we can find the time taken to cross the finish line. Using the equation of motion:
v = u + at
104 = 0 + 13.52t
Simplifying, we find:
t = 104/13.52 ≈ 7.69 seconds
To find the minimum coefficient of static friction, we can use the equation:
f_s = μ_s * N
where f_s is the force of static friction, μ_s is the coefficient of static friction, and N is the normal force.
Since the car is on a flat surface, the normal force is equal to the car's weight, which can be calculated using the formula:
N = m * g
where m is the mass of the car and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Unfortunately, we don't have the mass of the car. Therefore, we cannot find the minimum coefficient of static friction.
(b) If the acceleration is 30.0% smaller, the new acceleration would be 0.7 times the original acceleration (0.7 * 13.52 m/s^2).
Using the same equation of motion:
v = u + at
104 = 0 + (0.7 * 13.52)t
Simplifying, we find:
t = 104 / (0.7 * 13.52) ≈ 9.46 seconds
Therefore, if the acceleration is 30.0% smaller, it would take approximately 9.46 seconds to finish the race.
Note: The minimum coefficient of static friction remains unknown due to the absence of the mass of the car.