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'll break it down into smaller steps.
(a) First, let's find the time it takes for the car to come to a stop. We can use the equation for motion with constant acceleration:
v^2 = u^2 + 2as
Where:
v = final velocity (0 m/s, since the car comes to a stop)
u = initial velocity (33.4 m/s)
a = acceleration (-1.80 m/s^2)
s = displacement (unknown, distance the car travels)
Rearranging the equation, we have:
s = (v^2 - u^2) / (2a)
Substituting the given values:
s = (0^2 - 33.4^2) / (2 * -1.80)
Now we can calculate s, which represents the total distance the car travels before coming to a stop.
s = (-33.4^2) / (-3.60)
Now, to find the number of revolutions each tire makes, we need to find the distance traveled by each tire. Since the tire's circumference is given by 2πr (where r is the radius), we can calculate the number of revolutions using the formula:
number of revolutions = total distance / tire circumference
Since we already have the total distance, we need to calculate the tire circumference using the given radius:
tire circumference = 2π * 0.330 m
Finally, we can calculate the number of revolutions each tire makes:
number of revolutions = total distance / tire circumference
(b) To find the angular speed of the wheels when the car has traveled half the total distance, we first need to find the time it takes to travel half the total distance. We can calculate this by dividing the total distance by 2 and using the equation:
s = ut + (1/2)at^2
Where:
s = half the total distance (found in part a)
u = initial velocity (33.4 m/s)
a = acceleration (-1.80 m/s^2)
t = time
Rearranging the equation, we get a quadratic equation:
(1/2)at^2 + ut - s = 0
We can solve this quadratic equation to find t. After finding t, we can calculate the angular speed of the wheels using the formula:
angular speed = linear speed / tire radius
Where:
linear speed = u - at (using the equation of motion)
tire radius = 0.330 m
Now we have all the information to solve the problem and find the answers. Just substitute the calculated values into the formulas provided above.
To solve this problem, we can use the equations of motion for rotational motion.
For part (a):
1. First, let's 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 = final velocity (0 m/s, since the car comes to a stop)
u = initial velocity (33.4 m/s)
a = acceleration (-1.80 m/s^2, negative because it's decelerating)
s = distance (unknown)
Rearranging the equation:
s = (v^2 - u^2) / (2a)
s = (0^2 - 33.4^2) / (2 * -1.8)
s = (0 - 1115.56) / (-3.6)
s = 1015.56 / 3.6
s = 282.1 m
2. Now, let's find the number of revolutions each tire makes. The distance traveled by each tire can be calculated using the formula:
distance = 2πr * number of revolutions
Where:
distance = 282.1 m
r = radius of the tire (0.330 m)
number of revolutions = unknown
Rearranging the equation:
number of revolutions = distance / (2πr)
number of revolutions = 282.1 / (2 * π * 0.330)
number of revolutions ≈ 270.641
Therefore, each tire makes approximately 270.641 revolutions before the car comes to a stop.
For part (b):
1. First, let's find the total distance traveled by the car before it comes to a stop. Since the car has traveled half the total distance, the distance traveled can be calculated as:
total distance = 2 * d
Where:
d = unknown
total distance = 282.1 m (as previously calculated)
Rearranging the equation:
d = total distance / 2
d = 282.1 / 2
d = 141.05 m
2. Now, let's calculate the angular displacement of each tire. The angular displacement can be calculated using the formula:
angular displacement = distance / r
Where:
distance = 141.05 m (as previously calculated)
r = radius of the tire (0.330 m)
angular displacement = 141.05 / 0.330
angular displacement ≈ 427.88 radians
3. Finally, let's calculate the angular speed of the wheels when the car has traveled half the total distance. The angular speed can be calculated using the formula:
angular speed = initial angular velocity + angular acceleration * time
Where:
initial angular velocity = 0 rad/s (since the car starts from rest)
angular acceleration = -1.80 m/s^2 / r (negative to indicate deceleration)
time = unknown
Rearranging the equation, we can solve for time:
angular acceleration = (angular speed - initial angular velocity) / time
-1.80 / 0.330 = (angular speed - 0) / time
time = (0.330 * -1.80) / angular speed
Substituting the known values:
time ≈ -0.594 / 427.88
Therefore, the angular speed of the wheels when the car has traveled half the total distance is approximately -0.00139 rad/s.