A new car is tested on a 200-m-diameter track. If the car speeds up at a steady 1.5 m/s^2, how long after starting is the magnitude of its centripetal acceleration equal to the tangential acceleration?

Verified step by step guidance

1. Identify the given values: tangential acceleration (a_t) is 1.5 m/s^2, and the radius (r) of the circular track is 100 m (half of the diameter).
2. Understand that the magnitude of centripetal acceleration (a_c) is given by the formula a_c = \frac{v^2}{r}, where v is the velocity of the car.
3. Recognize that the velocity (v) of the car increases over time due to the tangential acceleration, following the formula v = a_t \times t, where t is the time elapsed.
4. Substitute the expression for v into the centripetal acceleration formula to get a_c = \frac{(a_t \times t)^2}{r}.
5. Set the magnitudes of centripetal and tangential accelerations equal to find the time t: a_t = \frac{(a_t \times t)^2}{r}. Solve this equation for t.

Key Concepts

Centripetal Acceleration

Centripetal acceleration is the acceleration directed towards the center of a circular path that keeps an object moving in that path. It is calculated using the formula a_c = v^2 / r, where v is the tangential speed and r is the radius of the circular path. In this scenario, as the car speeds up, its tangential speed increases, affecting the centripetal acceleration.

Tangential Acceleration

Tangential acceleration refers to the rate of change of the speed of an object moving along a circular path. It is constant in this case, given as 1.5 m/s². This acceleration is responsible for increasing the car's speed as it moves around the track, and it acts along the direction of the car's velocity.

Equating Accelerations

To solve the problem, we need to find the time when the magnitudes of centripetal and tangential accelerations are equal. This involves setting the expressions for both accelerations equal to each other and solving for the time variable. This requires understanding how both types of acceleration depend on the car's speed and the radius of the track.