Question

What is average velocity and speed of particle between t=0 and t=2pi if velocity is given by V=Sin (t).

Answer 1

The average velocity and speed of the particle are both 0.

Given:

Velocity, V = sin(t)

To Find:

The average velocity and speed of the particle are?

Solution:

We know that:

Average velocity = Total displacement / Total time and

Average speed = Total distance travelled / Total time

It is given that velocity is

V = sin(t)

Total displacement = Total distance travelled =  ${\int_{0}^{2\pi}{sin(t)\ dt}$

Total time = 2π - 0 = 2π

Average velocity = $\frac{\int_{0}^{2\pi}{sin(t)\ dt}}{\int_{0}^{2\pi}dt}$

                             = $\frac{\left[-cos(t)\right]_0^{2\pi}}{{t|}_0^{2\pi}}$

                             = $\frac{-cos2\pi - (-cos0)}{2\pi - 0}$

                             = $\frac{-1 - (-1)}{2\pi }$

                             = $\frac{-1 + 1}{2\pi }$

                             = $\frac{0}{2\pi }$

                             = 0

Therefore, the average velocity of the particle between t = 0 and t = 2π is 0.

The total distance travelled and total displacement is the same i.e., equal to 0. So, the average speed is equal to the average velocity.

That is why the average speed of the particle between t = 0 and t = 2π is also 0.

Hence, the average velocity and speed of the particle are both 0.

#SPJ4

Answer 2

EXPLANATION.

Average velocity and speed of particles between t = 0 to t = 2π.

Velocity is given by : v = sin(t).

We can write expression as,

$\sf \displaystyle Average \ velocity = \frac{\displaystyle \int_{0}^{2\pi} sin(t) dt}{\displaystyle \int_{0}^{2\pi} dt}$

$\sf \displaystyle Average \ velocity = \frac{\bigg[-cos(t)\bigg]_{0}^{2\pi}}{\bigg[t \bigg]_{0}^{2\pi}}$

As we know that,

In definite integrals first we put upper limits in the expression then we put lower limits in the expression, we get.

$\sf \displaystyle Average \ velocity = \frac{[- cos(2\pi) ] - [- cos(0)]}{(2\pi - 0)}$

$\sf \displaystyle Average \ velocity = \frac{-(1) - (-1)}{2 \pi}$

$\sf \displaystyle Average \ velocity = \frac{-1 + 1}{2 \pi}$

$\sf \displaystyle Average \ velocity = \frac{0}{2 \pi}$

$\sf \displaystyle Average \ velocity = 0$

∴ The average velocity of the particle is equal to 0.

As we know that,

Average speed = (Total distance covered)/(Total time taken).

Total Distance travelled = $\sf \displaystyle \int_{0}^{2 \pi} sin(t) dt$

$\sf \displaystyle \bigg[- cos(t) \bigg]_{0}^{2 \pi}$

$\sf \displaystyle [-cos(2 \pi)] - [- cos(0)]$

$\sf \displaystyle (-1) + (1) = 0$

Total distance travelled = 0.

Total time taken = Final time - initial time.

Total time taken = 2π - 0.

Total time taken = 2π.

Average speed = (0)/(2π).

Average speed = 0.

∴ The average speed of the particle is equal to 0.