Momentum in terms of force

STATEMENT:

Consider a spherical body of mass “$m$”, moving with initial velocity “$v_i$”. A force “$F$” acts on a body to produce acceleration “$a$”, therefore after time “$t$”  the body is found to be moving with final velocity “$v_f$”. Let, “$p_i$” & “$p_f$” respectively represents the initial and final momentum of the body and “$\Delta p$” represents the change in momentum of the body.

DERIVATION:

Since, mass “$m$” is constant. Hence, the change in momentum is due to the change in velocity of body.

Initial Momentum = $p_i$= $m\thinspace v_i$

Final Momentum = $p_f$ = $m \thinspace v_f$

Hence,

Change in momentum = $p_f - p_i$

$\Delta p = p_f -p_i$

$\Delta p = m \thinspace v_f - m \thinspace v_i$

$\Delta p = m( \thinspace v_f - \thinspace v_i)$

Divide by “$t$” on both sides:

$\boxed {\frac{\Delta p}{t} = m \frac{( \thinspace v_f - \thinspace v_i)}{t}}$ ………. (i)

We know that

$\boxed {a = \frac{( \thinspace v_f - \thinspace v_i)}{t}}$

Substitute value in equation (i):

Hence,

$\boxed {\frac{\Delta p}{t} = m \thinspace a}$ ………… (ii)

Since, $F = m \thinspace a$

Therefore, equation (ii) becomes:

$\boxed {\frac{\Delta p}{t} = F}$

CONCLUSION:

Since, rate of change in momentum = $F$

So, we can define the relation as follows:

“The rate of change in momentum of a body is equal to the force applied on the body”.