Q. 59 - Calculus

Question

Suppose a very strange particle moves back and forth along a straight path in such a way that its velocity after t seconds is given by $v\left(t\right)=\sin \left(0.1t^2\right)$, measured in feet per second. Consider positions left of the starting position to be in the negative direction and positions right of the starting position in the positive direction.

Part (a): Write down an expression for the position $s\left(t\right)$ of the very strange particle, measured in feet left or right of the starting position. Your expression for $s\left(t\right)$ will involve an integral.

Part (b): Use your answer to part (a) and a Riemann sum with $10$ rectangles to approximate the position of the particle after $10$ seconds of motion.

Part (c): Verify that the units in your calculation to part (b) make sense. Why is it clear that the Riemann sum will have units measured in feet?

Part (d): What happens to the velocity of the very strange particle after a long time? What does this mean about how the particle is moving after, say, about $100$ seconds?

Step-by-Step Solution

Answer

Part (a): The required expression is $s\left(t\right)={\int }_0^t \sin \left(0.1{\omega }^2\right) d\omega$.

Part (b): The position of the particle after $10$ seconds of motion will be about $2.411$feet to the right pf the starting position.

Part (c): Each rectangle has a height measured in feet per second and a width measured in seconds, thus an area that is measured in feet.

Part (d): From the expression $v\left(t\right)=\sin \left(0.1t^2\right)$.

The velocity oscillates faster and faster between $1,-1$ feet per second.

Part (a) Step 1. Given information.

Consider the given question,

Velocity of the particle,

$v\left(t\right)=\sin \left(0.1t^2\right) feet per second$

Part (a) Step 2. Write the expression for position s t .

The position of the particle is the integration of its velocity. So,

$s\left(t\right)={\int }_0^t \sin \left(0.1{\omega }^2\right) d\omega$

Where, $\omega$ is a dummy variable.

Part (b) Step 1. Determine the right sum.

Consider the function,

$f\left(t\right)=\sin \left(0.1t^2\right)$

The right sum defined for n rectangles on $\left[a,b\right]$ is $\sum _{k=1}^n f\left(x_k\right)△x$.

Where, $$\begin{gathered} △x=\frac{b-a}{n}, \\ {x}_k=a+k△x \end{gathered}$$

Now $△x$,

$$\begin{gathered} =\frac{10-0}{10} \\ =\frac{10}{10} \\ =1 \end{gathered}$$

Part (b) Step 2. Find the position after 10 seconds.

Consider the given question,

$$\begin{gathered} x_k=0+k\left(1\right) \\ =k \end{gathered}$$

The right sum is given below,

$$\begin{gathered} =\sum _{k=0}^{10} \sin \left(0.1k^2\right)1 \\ =\sin \left(0^2\right)+\sin \left(0.1\right)+\sin \left(0.4\right)+\sin \left(0.9\right)+\sin \left(1.6\right)+\sin \left(2.5\right)+\sin \left(3.6\right)+\sin \left(4.9\right)+\sin \left(6.4\right)+\sin \left(8.1\right)+\sin \left(1\right) \\ \approx 2.411 \end{gathered}$$

Part (c) Step 1. Explain how the Riemann sum will have units measured in feet.

Consider the given question,

Each rectangle has a height measured in feet per second and a width measured in seconds, thus an area that is measured in feet.

Part (d) Step 1. Explain how the particle will move after 100 seconds.

Consider the given graph,

$v\left(t\right)=\sin \left(0.1t^2\right)$

The velocity oscillates faster and faster between $1,-1$ feet per second.