A car is traveling with a velocity of 17 m/s on a straight horizontal highway. The wheels of the car have a radius of 48 cm. If the car then speeds up with an acceleration of 2 m/s² for 5 s, determine the number of revolutions of the wheels during this period.
the distance traveled is 17t + 1/2 at^2
with t=5 and a=2.
So, divide that by the circumference of the wheel = 2pi*48
Well, let's put on our clown shoes and calculate the number of revolutions!
First, we need to find the final velocity of the car after the acceleration. Using the formula v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time, we can plug in the values:
v = 17 m/s + (2 m/s²)(5 s)
v = 17 m/s + 10 m/s
v = 27 m/s
Now, let's calculate the distance traveled by the car during the acceleration period. Using the formula s = ut + (1/2)at², where s is the distance traveled, we can plug in the values:
s = (17 m/s)(5 s) + (1/2)(2 m/s²)(5 s)²
s = 85 m + (1/2)(2 m/s²)(25 s²)
s = 85 m + 25 m
s = 110 m
Phew, math can tire anyone out, even clowns!
Now, let's find the number of revolutions of the wheels. The distance traveled by one wheel in one revolution is equal to the circumference of the wheel, which can be calculated using the formula C = 2πr, where C is the circumference and r is the radius.
C = 2π(0.48 m)
C ≈ 3.022 m
So, the number of revolutions during the acceleration period is equal to the distance traveled divided by the circumference of one wheel:
Number of revolutions = 110 m / 3.022 m
Number of revolutions ≈ 36.41 revolutions
So, during this period, the wheels of the car would make approximately 36.41 revolutions. And by the end of it, they'll be dizzy enough to join the circus with us!
To determine the number of revolutions of the wheels, we need to find the distance covered by the car during the acceleration period.
Step 1: Convert the wheel radius to meters.
The radius of the wheels is given as 48 cm. To convert it to meters, we divide it by 100:
48 cm ÷ 100 = 0.48 m
Step 2: Find the initial velocity of the car.
The car is traveling with a velocity of 17 m/s.
Step 3: Find the final velocity of the car.
Using the equation for final velocity (v = u + at), where u is the initial velocity, a is the acceleration, and t is the time, we can calculate the final velocity.
v = u + at
v = 17 m/s + (2 m/s² * 5 s)
v = 17 m/s + 10 m/s²
v = 27 m/s
Step 4: Find the distance covered by the car during the acceleration.
Using the equation for distance covered (s = ut + (1/2)at²), we can calculate the distance covered.
s = ut + (1/2)at²
s = 17 m/s * 5 s + (1/2) * 2 m/s² * (5 s)²
s = 85 m + (1/2) * 2 m/s² * 25 s²
s = 85 m + 25 m
s = 110 m
Step 5: Calculate the circumference of the wheel.
Circumference = 2πr
Circumference = 2 * π * 0.48 m
Circumference ≈ 3.02 m
Step 6: Find the number of revolutions.
Number of revolutions = distance covered / circumference = 110 m / 3.02 m/rev ≈ 36.42 revolutions
Therefore, during the 5-second period with an acceleration of 2 m/s², the wheels of the car will make approximately 36.42 revolutions.
To determine the number of revolutions of the wheels during this period, we can follow these steps:
1. Calculate the initial angular velocity of the wheels.
- Angular velocity is the rate at which an object rotates. It is measured in radians per second (rad/s).
- The formula for angular velocity is: ω = v/r, where ω is the angular velocity, v is the linear velocity, and r is the radius of the wheels.
- Convert the radius of the wheels from centimeters to meters: r = 48 cm = 0.48 m.
- Substitute the values into the formula: ω = 17 m/s / 0.48 m = 35.42 rad/s.
2. Calculate the final angular velocity of the wheels.
- The final angular velocity can be found using the formula: ω = ω₀ + αt, where ω₀ is the initial angular velocity, α is the acceleration, and t is the time.
- Substitute the values into the formula: ω = 35.42 rad/s + (2 m/s²)(5 s) = 45.42 rad/s.
3. Calculate the change in angular velocity.
- The change in angular velocity (∆ω) is the difference between the final and initial angular velocities: ∆ω = ω - ω₀.
- Substitute the values into the formula: ∆ω = 45.42 rad/s - 35.42 rad/s = 10 rad/s.
4. Calculate the number of revolutions.
- One revolution is equal to the angle of one complete circle, which is 2π radians.
- Divide the change in angular velocity (∆ω) by the angular displacement for one revolution to find the number of revolutions: number of revolutions = ∆ω / (2π).
- Substitute the values into the formula: number of revolutions = 10 rad/s / (2π) ≈ 1.59 revolutions.
Therefore, during the 5-second period, the wheels of the car completed approximately 1.59 revolutions.