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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.

# Finding 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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