A car travels at a constant speed around a circular track whose radius is 2.5 km. The car goes once around the track in 430 s. What is the magnitude of the centripetal acceleration of the car?

m/s2

To find the magnitude of the centripetal acceleration of the car, we can use the formula:

Centripetal acceleration = (velocity^2) / radius

First, we need to find the velocity of the car.

Given:
Radius of the track (r) = 2.5 km
Time taken to complete one lap (t) = 430 s

To find the velocity, we can use the formula:

Velocity = (2 * π * r) / t

Substituting the given values:

Velocity = (2 * 3.14 * 2.5 km) / 430 s

Velocity = (15.7 km) / 430 s

Velocity ≈ 0.0365 km/s

To convert the velocity from km/s to m/s, we need to multiply by 1000:

Velocity ≈ 36.5 m/s

Now, we can substitute the values into the centripetal acceleration formula:

Centripetal acceleration = (36.5 m/s)^2 / 2.5 km

Since the centripetal acceleration is perpendicular to the path of the car, its direction is not specified. Therefore, we only need to find the magnitude.

Centripetal acceleration = (36.5 m/s)^2 / 2.5 km

Converting km to m, we get:

Centripetal acceleration = (36.5 m/s)^2 / 2500 m

Centripetal acceleration ≈ 0.532 m/s^2

Therefore, the magnitude of the centripetal acceleration of the car is approximately 0.532 m/s^2.

To find the magnitude of the centripetal acceleration, we can use the formula:

ac = v² / r

Where:
ac = centripetal acceleration
v = speed of the car
r = radius of the circular track

We are given the radius of the track, which is 2.5 km. However, we need to convert it to meters, since the unit for acceleration is m/s².

1 km = 1000 m

So, the radius (r) = 2.5 km * 1000 m/km = 2500 m

Next, we need to find the speed of the car. We are given that it takes 430 seconds for the car to go once around the track. To find the speed, we can use the formula:

v = 2πr / t

Where:
v = speed
π = 3.14 (approximately)
r = radius
t = time

Plugging in the values, we get:

v = 2 * 3.14 * 2500 m / 430 s

Simplifying, we have:

v = 14651.16 m / 430 s

v ≈ 34.07 m/s

Finally, we can calculate the magnitude of the centripetal acceleration by substituting the values into the formula:

ac = (34.07 m/s)² / 2500 m

ac ≈ 1.826 m/s²

Therefore, the magnitude of the centripetal acceleration of the car is approximately 1.826 m/s².