# The position of an object moving along a line is given by p(t) = t-cos(( pi )/2t) . What is the speed of the object at t = 3 ?

Dec 18, 2016

|v(t)|=|1-pi/2| ≈ 0.57 (units)

#### Explanation:

Speed is a scalar quantity having only magnitude (no direction). It refers to how fast an object is moving. On the other hand, velocity is a vector quantity, having both magnitude and direction. Velocity describes the rate of change of position of an object. For example, $40 \frac{m}{s}$ is a speed, but $40 \frac{m}{s}$ west is a velocity.

Velocity is the first derivative of position, so we can take the derivative of the given position function and plug in $t = 3$ to find the velocity. The speed will then be the magnitude of the velocity.

$p \left(t\right) = t - \cos \left(\frac{\pi}{2} t\right)$

$p ' \left(t\right) = v \left(t\right) = 1 + \frac{\pi}{2} \sin \left(\frac{\pi}{2} t\right)$

The velocity at $t = 3$ is calculated as

$v \left(3\right) = 1 + \frac{\pi}{2} \sin \left(\frac{3 \pi}{2}\right)$

$v \left(3\right) = 1 - \frac{\pi}{2}$

And then the speed is simply the magnitude of this result, such as that speed= $| v \left(t\right) |$

|v(t)|=|1-pi/2| ≈ 0.57 (units)