A car starts from rest on a curve with a radius of 130 m and accelerates at 1.80 m/s^2. How many revolutions will the car have gone through when the magnitude of its total acceleration is 3.50 m/s^2?

Question

A car starts from rest on a curve with a radius of 130 m and accelerates at 1.80 m/s^2. How many revolutions will the car have gone through when the magnitude of its total acceleration is 3.50 m/s^2?

Answer 1

To answer this question, we need to break it down into steps:

Step 1: Find the magnitude of the car's total acceleration.
Step 2: Use the magnitude of the total acceleration to find the car's speed at that point.
Step 3: Use the car's speed to calculate the number of revolutions it has gone through.

Let's work through each step:

Step 1: Find the magnitude of the car's total acceleration.
The magnitude of the car's total acceleration is given as 3.50 m/s^2.

Step 2: Use the magnitude of the total acceleration to find the car's speed at that point.
We know that the car started from rest, so its initial velocity is 0 m/s. We can use the following formula to find the car's final velocity:

v^2 = u^2 + 2as

where:
v = final velocity
u = initial velocity
a = acceleration
s = displacement

In this case, the initial velocity is 0 m/s, the acceleration is 3.50 m/s^2, and the radius of the curve (displacement) is 130 m. Plugging these values into the formula, we get:

v^2 = 0^2 + 2 * 3.50 * 130

v^2 = 2 * 3.50 * 130

v^2 = 910

Taking the square root of both sides, we get:

v = √(910)

v ≈ 30.18 m/s

Step 3: Use the car's speed to calculate the number of revolutions it has gone through.
The speed of the car at this point is approximately 30.18 m/s. To calculate the number of revolutions, we need to convert the speed from meters per second to meters per revolution.

The formula to convert meters per second to meters per revolution is:

meters per revolution = circumference of the circle / speed

The circumference of the circle can be calculated using the formula:

circumference = 2π * radius

In this case, the radius is 130 m, so the circumference is:

circumference = 2π * 130

Now we can find the number of revolutions using the formula:

number of revolutions = speed / meters per revolution

number of revolutions = 30.18 / (2π * 130)

number of revolutions ≈ 0.046 revolutions

Therefore, the car will have gone through approximately 0.046 revolutions when the magnitude of its total acceleration is 3.50 m/s^2.