An automobile whose speed is increasing at a rate of 0.800 m/s2 travels along a circular road of radius 10.0 m.

(a) when the instantaneous speed of the automobile is 3.00 m/s, find the tangential acceleration component.

(b) what is the centripetal acceleration component?

(c) determine the magnitude and direction of the total acceleration.

(a)

The tangential acceleration component of the car is simply equal to the change of the tangential speed divided by the time taken:

This rate of change is already given by the problem, 0.800 m/s^2, so the tangential acceleration of the car is

(b)

The centripetal acceleration component is given by

where

v is the tangential speed

r is the radius of the trajectory

When the speed is v = 3.00 m/s, the centripetal acceleration is (the radius is r = 10.0 m):

(c)

The centripetal acceleration and the tangential acceleration are perpendicular to each other, so the magnitude of the total acceleration can be found by using Pythagorean's theorem:

and the direction is given by:

where the angle is measured with respect to the direction of the tangential acceleration.

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