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# Initial Velocity

Top
 Sub Topics When the motion of a body is being considered, we have following factors to explain the motion, initial velocity, final velocity, acceleration, displacement and time interval. Here the initial velocity of a substance is the velocity the body will be moving at time $t = 0$. By the known value of initial velocity one can calculate other parameters related to the motion of a body. In this section we will learn more about initial velocity.

## What is Initial Velocity?

When an object is set in to motion, the velocity with which it moves in starting is called Initial Velocity. Normally when an object is set into motion from rest then the initial velocity will be equal to zero but it depends on various other factors like initial impact, state of motion used a reference for calculation due to which it may not be zero.
For example when a ball is set into motion from rest at $t=0$ (time) and subsequent observations of final velocity are taken afterwards. But suppose in other reference set the observations are started to be taken at $t=1$ sec then the initial velocity for second set of observation will not be zero rather it will be equal to some value.

## Initial Velocity Formula

There are total three set of equations which are used to calculate the initial velocity or Initial velocity is used in these equations and is used to calculate any of the other component

$v = u + at$

$s$ = $ut$ + $\frac{1}{2}$ $at^{2}$

$s$ = $\frac{1}{2}(v+u)t$

$v^{2} = u^{2} + 2as$

Where,
$v$ = Final velocity
$u$ = Initial velocity
$t$ = The time
$a$ = Acceleration
$s$ = Displacement

In case of deceleration a will be negative (i.e $a<0$) and subsequent final velocity will be lesser then initial velocity (i.e $v<u$) or else if acceleration is their (i.e $a>0$) then final velocity will be higher then initial velocity (i.e $v>u$).

## Initial Velocity and Final Velocity

The initial velocity of a substance is the velocity it will be moving at time $t = 0$. The final velocity of the substance is the velocity of the substance that will be moving at time t.

Both final and initial velocity can be zero at certain situation.

If a car starts from rest, its initial velocity is zero. If a projectile is tossed into the space, its initial velocity will be more than zero. If a car stops after applying the brake, the initial velocity will be more than zero, but the final velocity will be zero.

## Finding Initial Velocity

The following are the initial velocity problems.

### Solved Examples

Question 1: A motor bike acquires a velocity of 72 km/h in 10s starting from the rest. Calculate the acceleration?
Solution:

Given :

Initial velocity $u$ = $0$

Final velocity $v$ = $72$ km/h

= $\frac{72 \times 1000}{60 \times 60}$ = $20$ m/s

Time taken = $10$s.

Acceleration $a$ = $\frac{v-u}{t}$

= $\frac{20-0}{10}$

= $2$ m/s2.

Question 2: An arrow is thrown upwards in the air from the ground. The maximum height of the arrow is $h$ = $16 t^{2} + 64t + 6$. Find the initial velocity of the arrow?
Solution:

Given the maximum height is $h$ = $16t^{2} + 64t + 6$

We want to find the initial velocity of the arrow.

$V$ = $\frac{dh}{dt}$ = $\frac{d}{dt}$ $(16t^{2} + 64t + 6)$

$V$ = $\frac{dh}{dt}$ = $32t + 64$

To find the initial velocity we have to plug time $t = 0$.

$V$ = $32 (0) + 64$

Initial velocity is $V = 64$.

Question 3: A ball is thrown upwards. It reaches a height of 10 m and then starts falling down. Can you calculate the initial velocity with which the ball was thrown upwards?
Solution:

Given,
Distance $S$ = $10$ m
Acceleration due to gravity $g$ = $9.8$ m/s2
Final velocity $v = 0$

We have
$v^{2}$ = $u^{2} + 2as$
by rearranging this
we have  $s$ = $\frac{u^{2}}{2a}$
Putting the values in the equation,
we get
$10$ = $\frac{u^{2}}{2 \times 9.8}$
$u^{2}$ = $10 \times 2 \times 9.8$ = $196$
$u$ = $14$ m/s