The driver of a car traveling at 25.0 m/s applies the brakes and undergoes a constant negative acceleration of magnitude 2.50 m/s2. How many revolutions does each tire make before the car comes to a stop, assuming that the car does not skid and that the tires have radii of 0.27 m?
V^2 = Vo^2 + 2a*d = 0
d = -Vo^2/2a = -(25^2)/-5 = 125 m.
Circumference = pi * 2r = 3.14 * 0.54 =
1.70 m.
125m * 1rev/1.70m = 73.5 Revolutions.
Why did the car go to school? Because it wanted to make some revolutions!
To find the number of revolutions each tire makes before the car comes to a stop, we can use the formula for angular displacement:
θ = (v₀t + (1/2)at²) / r
Where:
θ is the angular displacement
v₀ is the initial velocity (25.0 m/s)
t is the time taken for the car to stop
a is the acceleration (2.50 m/s²)
r is the radius of the tire (0.27 m)
First, let's find the time it takes for the car to stop. We can use the equation:
v = v₀ + at
0 = 25.0 m/s + (-2.50 m/s²)t
Rearranging the equation, we get:
t = -25.0 m/s / -2.50 m/s²
t = 10.0 seconds
Now we can plug this time into the formula for angular displacement:
θ = (v₀t + (1/2)at²) / r
θ = (25.0 m/s * 10.0 s + (1/2)(-2.50 m/s²)(10.0 s)²) / 0.27 m
Calculating this, we find:
θ ≈ 1110 radians
But we want to find the number of revolutions. So, let's convert radians to revolutions:
1 revolution = 2π radians
θ = 1110 radians * (1 revolution / 2π radians)
θ ≈ 176.78 revolutions
Therefore, each tire makes approximately 176.78 revolutions before the car comes to a stop.
To find the number of revolutions each tire makes before the car comes to a stop, we can follow these steps:
Step 1: Calculate 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 (25.0 m/s)
- a = acceleration (-2.50 m/s^2 because it's in the opposite direction)
- s = displacement (unknown, to be calculated)
Rearranging the equation to solve for s:
s = (v^2 - u^2) / (2a)
s = (0^2 - 25.0^2) / (2 * -2.50)
s = 625.0 / -5.00
s = -125.0 m (negative sign indicates it is in the opposite direction)
Step 2: Calculate the distance traveled by each tire.
The distance traveled by each tire is equal to the circumference of a circle with a radius of 0.27 m. The formula for the circumference is given by: C = 2πr.
C = 2 * π * 0.27
C = 1.70 m (approximately)
Step 3: Calculate the number of revolutions for each tire.
We can calculate the number of revolutions using the formula:
Number of revolutions = distance traveled / circumference of the tire.
Number of revolutions = -125.0 / 1.70
Number of revolutions ≈ -73.53
So, each tire makes approximately 73.53 revolutions before the car comes to a stop.
To find the number of revolutions made by each tire before the car comes to a stop, we can first calculate the time it takes for the car to come to a stop using the given information.
We are given:
Initial velocity (v0) = 25.0 m/s
Acceleration (a) = -2.50 m/s² (negative because it is decelerating)
To find the time it takes for the car to come to a stop, we can use the equation of motion:
v = v0 + at
where:
v = final velocity (which is 0 m/s since the car comes to a stop)
v0 = initial velocity
a = acceleration
t = time
We substitute the values into the equation:
0 = 25.0 m/s + (-2.50 m/s²) * t
Simplifying the equation:
-25.0 m/s = -2.50 m/s² * t
Divide both sides by -2.50 m/s² to isolate t:
t = (25.0 m/s) / (2.50 m/s²) = 10.0 s
Now that we know the time it takes for the car to stop, we can find the distance traveled by each tire before the car stops. Since the car is not skidding, the distance traveled is the same as the distance covered by the center of the tires.
The distance covered by a tire can be calculated using the formula:
d = vt
where:
d = distance
v = velocity
t = time
For one complete revolution, the tire travels a distance equal to its circumference. The circumference of a tire can be found using the formula:
C = 2πr
where:
C = circumference
r = radius of the tire
Now, substituting the known values into the formula, we have:
C = 2π (0.27 m)
Calculating the circumference of each tire:
C = 2 * 3.14 * 0.27 m ≈ 1.70 m
The distance covered by each tire before the car stops is the circumference multiplied by the number of revolutions. Let's denote the number of revolutions as N. So,
d = N * C
Now, we need to find the number of revolutions made by each tire. We can rearrange the formula to solve for N:
N = d / C
Substituting the known values, we have:
N = (vt) / C
Substituting the values:
N = (25.0 m/s) * (10.0 s) / (1.70 m)
Calculating the number of revolutions for each tire:
N ≈ 147.05 revolutions
Therefore, each tire makes approximately 147.05 revolutions before the car comes to a stop.