A car initially traveling at 32.2 m/s undergoes a constant negative acceleration of magnitude 1.70 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 0.350 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 need to use the equations of linear motion and connect them with rotational motion.
(a) To find the number of revolutions each tire makes before the car comes to a stop, we can use the equations of linear motion together with the equation that relates linear motion to rotational motion.
1. First, find the time it takes for the car to come to a stop:
We can use the equation of linear motion: v = u + at, where
v = final velocity (0 m/s since the car comes to a stop)
u = initial velocity (32.2 m/s)
a = acceleration (-1.70 m/s^2)
t = time
Rearranging the equation to solve for t, we have:
t = (v - u) / a
t = (0 - 32.2) / -1.70
t ≈ 18.94 s
2. Next, we need to find the distance the car traveled before coming to a stop:
Using the equation of linear motion: s = ut + (1/2)at^2, where
s = distance traveled
u = initial velocity (32.2 m/s)
t = time (18.94 s)
a = acceleration (-1.70 m/s^2)
Substituting the values into the equation, we have:
s = (32.2 * 18.94) + (0.5 * -1.70 * (18.94)^2)
s ≈ 304.28 m
3. Finally, we can calculate the number of revolutions each tire makes:
The distance traveled by each tire is equal to the circumference of the tire times the number of revolutions.
The circumference of a tire is given by: C = 2πr, where r is the radius (0.350 m).
Number of revolutions = s / (C * 2π)
Number of revolutions = 304.28 / (2π * 0.350 * 2π)
Calculating this value will give you the answer in revolutions.
(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 to travel half the total distance and then calculate the angular speed.
1. Half of the total distance:
Half of the total distance is s/2, which we already calculated as 304.28 m.
2. Find the time it takes to travel half the total distance:
We can use the equation of linear motion: s = ut + (1/2)at^2, where
s = distance traveled (s/2)
u = initial velocity (32.2 m/s)
t = time (unknown)
a = acceleration (-1.70 m/s^2)
Substituting the values into the equation, we have:
s/2 = (32.2 * t) + (0.5 * -1.70 * t^2)
Solving this quadratic equation will give us the time it takes to travel half the total distance.
3. Calculate the angular speed:
Angular speed is given by the formula: ω = v/r, where
ω = angular speed
v = linear velocity
r = radius of the wheels
Since the linear velocity is equal to the circumference of the wheel times the angular speed, we can write: v = 2πr * ω.
Substituting the values into the equation, we have:
2πr * ω = (32.2 * t)
Solving this equation for ω will give you the angular speed in radians per second.