# The position of an object moving along a line is given by p(t) = t - tsin(( pi )/4t) . What is the speed of the object at t = 7 ?

Feb 24, 2018

$- 2.18 \text{m/s}$ is its velocity, and $2.18 \text{m/s}$ is its speed.

#### Explanation:

We have the equation $p \left(t\right) = t - t \sin \left(\frac{\pi}{4} t\right)$

Since the derivative of position is velocity, or $p ' \left(t\right) = v \left(t\right)$, we must calculate:

$\frac{d}{\mathrm{dt}} \left(t - t \sin \left(\frac{\pi}{4} t\right)\right)$

According to the difference rule, we can write:

$\frac{d}{\mathrm{dt}} t - \frac{d}{\mathrm{dt}} \left(t \sin \left(\frac{\pi}{4} t\right)\right)$

Since $\frac{d}{\mathrm{dt}} t = 1$, this means:

$1 - \frac{d}{\mathrm{dt}} \left(t \sin \left(\frac{\pi}{4} t\right)\right)$

According to the product rule, $\left(f \cdot g\right) ' = f ' g + f g '$.

Here, $f = t$ and $g = \sin \left(\frac{\pi t}{4}\right)$

$1 - \left(\frac{d}{\mathrm{dt}} t \cdot \sin \left(\frac{\pi t}{4}\right) + t \cdot \frac{d}{\mathrm{dt}} \left(\sin \left(\frac{\pi t}{4}\right)\right)\right)$

$1 - \left(1 \cdot \sin \left(\frac{\pi t}{4}\right) + t \cdot \frac{d}{\mathrm{dt}} \left(\sin \left(\frac{\pi t}{4}\right)\right)\right)$

We must solve for $\frac{d}{\mathrm{dt}} \left(\sin \left(\frac{\pi t}{4}\right)\right)$

Use the chain rule:

$\frac{d}{\mathrm{dx}} \sin \left(x\right) \cdot \frac{d}{\mathrm{dt}} \left(\frac{\pi t}{4}\right)$, where $x = \frac{\pi t}{4}$.

$= \cos \left(x\right) \cdot \frac{\pi}{4}$

$= \cos \left(\frac{\pi t}{4}\right) \frac{\pi}{4}$

Now we have:

$1 - \left(\sin \left(\frac{\pi t}{4}\right) + \cos \left(\frac{\pi t}{4}\right) \frac{\pi}{4} t\right)$

$1 - \left(\sin \left(\frac{\pi t}{4}\right) + \frac{\pi t \cos \left(\frac{\pi t}{4}\right)}{4}\right)$

$1 - \sin \left(\frac{\pi t}{4}\right) - \frac{\pi t \cos \left(\frac{\pi t}{4}\right)}{4}$

That's $v \left(t\right)$.

So $v \left(t\right) = 1 - \sin \left(\frac{\pi t}{4}\right) - \frac{\pi t \cos \left(\frac{\pi t}{4}\right)}{4}$

Therefore, $v \left(7\right) = 1 - \sin \left(\frac{7 \pi}{4}\right) - \frac{7 \pi \cos \left(\frac{7 \pi}{4}\right)}{4}$

$v \left(7\right) = - 2.18 \text{m/s}$, or $2.18 \text{m/s}$ in terms of speed.