A sphere is fired downwards into a medium with an initial speed of 27 m/s. If it experiences a deceleration of a = (-6t) m/s^2 where t is in seconds, determine the distance traveled before it stops.
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When acceleration is given with respect to time, we can write it as:
solving for dv yields:dv=a(t)\,dt
We can now take the integral of both sides to find the time when the sphere stops.
Substituting our acceleration equation gives us:
When the sphere comes to a stop, v = 0. Thus, the time is:
To find the distance traveled, remember that:
Rearranging the equation we found for the time so that velocity is isolated:
Substitute this value into our equation and take the integral:
When t = 3 seconds, we have: