A particle moves along the \(x\)-axis with velocity
\[v(t)=30t^2-90t+60\]
at time \(t\). Find the total distance covered between time \(t=0\) and time \(t=5.\)

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A train starts from station \(A\) and arrives at station \(B\) in \(6\) minutes. If the velocity of the train in \(\text{m/min}\) is \[v(t) = 24t^2(6-t),\] what is the distance between the two stations \(A\) and \(B?\)

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Sam throws a ball straight upward at a speed of \(30 \text{ m/s}\) from the edge of a cliff \(19 \text{ m}\) high, as shown in the above diagram. If the the velocity of the ball \(t\) seconds after the ball leaves his hand is \(v=30-10t\) (in \(\text{m/s}\)), what is the distance of the ball from the ground \(3\) seconds after the ball leaves his hand?

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Two points \(P\) and \(Q\) start at the origin and move from the origin at the same time along the \(x\)-axis. At time \(t>0\), \(P\) and \(Q\) have velocities of \(\sin (\pi t)\) and \(2\sin (2\pi t)\), respectively. If \(P\) and \(Q\) first meet again after \(\frac{m}{n}\) seconds, where \(m\) and \(n\) are positive co-prime integers, what is the value of \(m+n\)?

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