A car is traveling counterclockwise along a circular bend in the road with an effective radius of 200 m. At a certain instant, the car’s speed is 20 m/s, but it is slowing down at a rate of 1.0 m/s^2. What is the magnitude of the car’s acceleration?
To find the magnitude of the car's acceleration, we need to consider two components: the tangential acceleration and the centripetal acceleration.
The tangential acceleration is in the direction of the velocity change. Since the car is slowing down, the tangential acceleration is negative.
The centripetal acceleration, on the other hand, is directed towards the center of the circular bend.
Given that the car's speed is 20 m/s and it is slowing down at a rate of 1.0 m/s^2, we can calculate the magnitude of the car's acceleration using the following steps:
1. Calculate the tangential acceleration:
Tangential acceleration = -1.0 m/s^2 (negative because it is slowing down)
2. Calculate the centripetal acceleration:
The centripetal acceleration can be obtained using the following formula:
Centripetal acceleration = (velocity^2) / radius
Substituting the values we have:
Centripetal acceleration = (20 m/s)^2 / 200 m
Centripetal acceleration = 400 m^2/s^2 / 200 m
Centripetal acceleration = 2 m/s^2
3. Calculate the total acceleration:
Since acceleration is a vector quantity, we need to find the vector sum of the tangential acceleration and the centripetal acceleration.
Total acceleration = sqrt((tangential acceleration)^2 + (centripetal acceleration)^2)
Total acceleration = sqrt((-1.0 m/s^2)^2 + (2 m/s^2)^2)
Total acceleration = sqrt(1.0 m^2/s^4 + 4 m^2/s^4)
Total acceleration = sqrt(5 m^2/s^4)
Therefore, the magnitude of the car's acceleration is sqrt(5) m/s^2, which is approximately 2.24 m/s^2.