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Displacement, Velocity, Acceleration

Derivatives are rates of change, and in the physical world that means things like velocity and acceleration. In fact, studying these quantities played a major role in the invention of Calculus.

Displacement Given Velocity

         

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 above is the velocity-time graph of a runner. How far does this runner travel for \(16 \) seconds?

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 \(B?\)

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?

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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