A race car starts from rest on a circular track of radius 565 m. The car's speed increases at the constant rate of 0.520 m/s2. At the point where the magnitudes of the centripetal and tangential accelerations are equal, find the following.
(a) the speed of the race car
m/s
(b) the distance traveled
m
( c) the elapsed time
s
a) V^2/R = 0.520 m/s^2
V = 17.141 m/s
c) t = V/a = 32.963 s
b) S = (V/2)*t = 282.51 m
(measured along a circular arc)
It will have completed 0.499 radians of the track, or about 28.6 degrees
To find the speed of the race car, we need to find the point where the magnitudes of the centripetal and tangential accelerations are equal. At this point, the tangential acceleration is equal to the centripetal acceleration.
The tangential acceleration can be calculated using the equation:
at = α × r
where at is the tangential acceleration, α is the angular acceleration, and r is the radius of the circular track.
Given that the speed of the car increases at a constant rate, the angular acceleration is equal to the rate of change of speed divided by the radius:
α = Δv / Δt
where Δv is the change in speed and Δt is the change in time.
The centripetal acceleration (ac) is given by:
ac = (v^2) / r
where v is the speed of the car.
Since the tangential acceleration is equal to the centripetal acceleration, we have:
at = ac
Substituting the equations for at and ac:
α × r = (v^2) / r
Rearranging the equation:
v^2 = α × r^2
Taking the square root of both sides:
v = √(α × r^2)
Let's calculate the speed of the race car:
α = 0.520 m/s^2
r = 565 m
v = √(0.520 m/s^2 × (565 m)^2)
v ≈ 76.3 m/s
(a) The speed of the race car is approximately 76.3 m/s.
To find the distance traveled by the race car, we need to calculate the time it takes to reach this speed.
The formula to calculate the final velocity of an object undergoing constant acceleration is:
v = u + at
where v is the final velocity, u is the initial velocity (which is 0 since the car starts from rest), a is the acceleration, and t is the time.
In this case, the final velocity v is the speed of the race car, a is the tangential acceleration, and t is the time taken to reach this speed.
Rearranging the equation, we have:
t = (v - u) / a
Substituting the values we have:
u = 0 m/s
v ≈ 76.3 m/s
a = 0.520 m/s^2
t = (76.3 m/s - 0 m/s) / 0.520 m/s^2
t ≈ 147.12 s
(b) The distance traveled by the race car can be calculated using the equation:
s = ut + 0.5at^2
where s is the distance traveled, u is the initial velocity, a is the acceleration, and t is the time.
In this case, the initial velocity u is 0 m/s, the acceleration a is the tangential acceleration, and the time t is approximately 147.12 s.
Substituting the values we have:
u = 0 m/s
a = 0.520 m/s^2
t ≈ 147.12 s
s = 0 × 147.12 + 0.5 × 0.520 m/s^2 × (147.12 s)^2
s ≈ 11,820 m
(c) The elapsed time is approximately 147.12 s.