A race car starts from rest on a circular track of radius 445 m. The car's speed increases at the constant rate of 0.380 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
(b) the distance traveled
( c) the elapsed time
To find the answer to these questions, we need to understand the relationship between centripetal acceleration, tangential acceleration, speed, and radius in circular motion.
In circular motion, the centripetal acceleration (ac) is given by the formula:
ac = v^2 / r
where v is the speed of the object and r is the radius of the circular track.
The tangential acceleration (at) is given by the formula:
at = dv / dt
where dv is the change in velocity and dt is the change in time.
We are given that at a certain point in the motion, the magnitudes of the centripetal and tangential accelerations are equal. So we can set the equations for ac and at equal to each other:
v^2 / r = dv / dt
Now, let's solve the problem step by step:
(a) To find the speed of the race car:
Rearranging the equation, we have:
v^2 = ac * r
Substituting the given values, we have:
v^2 = 0.380 m/s^2 * 445 m
v^2 = 169.9 m^2/s^2
Taking the square root of both sides, we get:
v = √(169.9) m/s
v ≈ 13.03 m/s
Therefore, the speed of the race car is approximately 13.03 m/s.
(b) To find the distance traveled:
We can use the formula for distance traveled on a circular track:
d = 2πr
Substituting the given value of the radius, we have:
d = 2 * π * 445 m
d ≈ 2799.6 m
Therefore, the distance traveled by the race car is approximately 2799.6 m.
(c) To find the elapsed time:
We know that the tangential acceleration (at) is equal to dv / dt. Rearranging this equation, we have:
dt = dv / at
We are given that the tangential acceleration (at) is constant, so we can use the average speed formula to find dv:
dv = at * t
Substituting the given value of the tangential acceleration, we have:
dv = 0.380 m/s^2 * t
Substituting the value of v that we found in part (a), we have:
dv = 13.03 m/s * t
Now we can substitute these values into the equation for dt:
dt = (13.03 m/s * t) / 0.380 m/s^2
Simplifying this equation, we get:
dt = 34.29 t
Now we can solve for t by dividing both sides of the equation by 34.29:
t = dt / 34.29
Therefore, the elapsed time is equal to dt divided by 34.29.