Displacement, Velocity, Acceleration
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 \(t=5.\)
The figure’s vertical scaling is set by \( v_{s} = 8.0 \text{ m/s.} \)
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 \(B?\)
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 \(m+n\)?