Circling Boat

Question:

Starting from rest a motorboat travels around a circular path of radius r=50 meters at a speed v=0.2*t² meters per second. Determine the magnitude of the boat's velocity and acceleration at time t=3 seconds.

Answer:

Part of the boats acceleration, the tangential part, will go toward increasing speed. Part of it, the radial part, will go toward maintaining the circular path. The boat is actually going to have to point in toward the center of the circle a bit to stay on the circle.

In circular motion, the radial acceleration is v²/r where v is the tangential velocity. We know that v is 0.2*t² so radial acceleration as a function of time is ar=.04*t⁴/50. The tangential acceleration is dv/dt=0.4t. The total acceleration magnitude is the square root of the sum of the squares of the tangential and radial components. or ((.04*t⁴/50)²+(0.4t)²)^.5. Plug t=3 into this to get acceleration at time 3. The speed is just the magnitude of the tangential velocity.

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