A 605-kg racing car completes one lap in 14.3 s around a circular track with a radius of 40.0 m. The car moves at constant speed.

a) What is the acceleration of the car?

(b) What force must the track exert on the tires to produce this acceleration?

after 3 minutes

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

\[a = \frac{v^2}{r}\]

where \(a\) is the acceleration, \(v\) is the speed, and \(r\) is the radius of the circular track.

We are given the radius of the track, \(r = 40.0\) m, and the time taken to complete one lap, \(t = 14.3\) s. To find the speed of the car, we can use the formula:

\[v = \frac{2\pi r}{t}\]

where \(v\) is the speed, \(r\) is the radius, and \(t\) is the time.

Substituting the given values, we get:

\[v = \frac{2\pi \cdot 40.0}{14.3} = 27.986 \, \text{m/s}\]

Now, substituting the value of speed into the acceleration formula, we find:

\[a = \frac{(27.986)^2}{40.0} = 19.5 \, \text{m/s}^2\]

Therefore, the acceleration of the car is \(19.5 \, \text{m/s}^2\).

To determine the force exerted by the track on the tires, we can use Newton's second law of motion:

\[F = m \cdot a\]

where \(F\) is the force, \(m\) is the mass, and \(a\) is the acceleration.

The mass of the racing car is given as \(m = 605\) kg. Substituting the values into the equation, we find:

\[F = 605 \cdot 19.5 = 11,797.5 \, \text{N}\]

Therefore, the force that the track must exert on the tires to produce this acceleration is \(11,797.5 \, \text{N}\).

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

a = v^2 / r

where:
a = acceleration
v = velocity
r = radius of the circular track

In this case, we are given that the car completes one lap in 14.3 seconds, which means it travels a distance equal to the circumference of the circular track in that time. The circumference can be calculated using the formula:

C = 2πr

Now, we have the time and the distance travelled, so we can find the velocity:

v = d / t

where:
v = velocity
d = distance
t = time

We know that the distance is equal to the circumference of the track, so:

d = C

Substituting this into the equation for velocity, we get:

v = C / t

Substituting the value of the circumference, we have:

v = (2πr) / t

Now, we have the velocity and the radius, so we can calculate the acceleration using the formula for centripetal acceleration:

a = v^2 / r

Now, let's substitute the given values into the equations to find the answers.

a) To find the acceleration of the car:
1. Calculate the circumference of the circular track using the formula:
C = 2πr
C = 2π(40.0 m)
C ≈ 251.33 m

2. Calculate the velocity of the car using the formula:
v = C / t
v = 251.33 m / 14.3 s
v ≈ 17.57 m/s

3. Substitute the values of velocity and radius into the formula for centripetal acceleration:
a = v^2 / r
a = (17.57 m/s)^2 / 40.0 m
a ≈ 7.68 m/s^2

Therefore, the acceleration of the car is approximately 7.68 m/s^2.

b) To find the force that the track must exert on the tires:
We know that force (F) is related to mass (m) and acceleration (a) by the formula:

F = ma

Substituting the given values:

F = (605 kg)(7.68 m/s^2)
F ≈ 4648.4 N

Therefore, the track must exert a force of approximately 4648.4 Newtons on the tires to produce this acceleration.