A car initially traveling at 33.4 m/s undergoes a constant negative acceleration of magnitude 1.80 m/s2 after its brakes are applied.
(a) How many revolutions does each tire make before the car comes to a stop, assuming the car does not skid and the tires have radii of 0.330 m?
___rev
(b) What is the angular speed of the wheels when the car has traveled half the total distance?
___rad/s
To solve this problem, we can use the equations of motion and the relationships between linear and angular quantities.
(a) Let's first find the time it takes for the car to come to a stop. We can use the equation of motion:
v^2 = u^2 + 2as
where v is the final velocity (0 m/s in this case as the car comes to a stop), u is the initial velocity (33.4 m/s), a is the acceleration (-1.80 m/s^2), and s is the distance. Solving for s, we have:
s = (v^2 - u^2) / (2a)
= (0 - (33.4)^2) / (2(-1.8))
= 616.78 m
Next, let's find the total distance traveled by the car. Since the car is coming to a stop, this distance is equal to the initial distance covered before coming to a stop.
The distance traveled by each wheel is equal to the circumference of the wheel multiplied by the number of revolutions made. The circumference of the wheel is given by:
circumference = 2πr
where r is the radius of the wheel (0.330 m). Therefore, the distance traveled by each wheel is:
distance_per_revolution = 2πr
To find the number of revolutions, we can divide the total distance traveled by each wheel by the distance per revolution:
number_of_revolutions = total_distance / distance_per_revolution
= 616.78 m / (2π * 0.330 m)
≈ 294.79 revolutions
Therefore, each tire makes approximately 294.79 revolutions before the car comes to a stop.
(b) To find the angular speed of the wheels when the car has traveled half the total distance, we need to find the time it takes for the car to cover half the total distance. Let's call this time t.
Since the car is undergoing uniform acceleration, we can use the equation of motion:
s = ut + (1/2)at^2
where s is the distance covered (616.78 m / 2 = 308.39 m), u is the initial velocity (33.4 m/s), a is the acceleration (-1.80 m/s^2), and t is the time. Solving for t, we have:
t = (-u ± √(u^2 - 2as)) / a
= (-33.4 ± √(33.4^2 - 2(-1.80)(308.39))) / (-1.80)
Taking the positive value, we get:
t ≈ 9.78 s
Next, let's find the angular speed of the wheels. The angular speed, ω, is given by:
ω = v / r
where v is the linear velocity and r is the radius of the wheel.
The linear velocity, v, can be calculated using the equation:
v = u + at
where u is the initial velocity (33.4 m/s) and a is the acceleration (-1.80 m/s^2). Substituting the values into the equation, we have:
v = 33.4 - 1.80 * t
= 33.4 - 1.80 * 9.78
≈ 16.58 m/s
Substituting the values into the equation for angular speed, we have:
ω = 16.58 m/s / 0.330 m
≈ 50.24 rad/s
Therefore, the angular speed of the wheels when the car has traveled half the total distance is approximately 50.24 rad/s.