Circling Boat

## Question:

Starting from rest a motorboat travels around a circular path of
radius r=50 meters at a speed
v=0.2*t^{2} 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 v^{2}/r
where v is the tangential velocity. We know that v is
0.2*t^{2} so radial acceleration as a function of time is
ar=.04*t^{4}/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*t^{4}/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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