Impulse – Momentum Theorem and Impact

Section I – Impulse – Momentum Theorem

Linear Momentum:

Linear momentum = mass * velocity = m * v

Momentum is vector quantity and its SI unit is kg.m/sec.

Angular Momentum

Angular momentum = I . w
Angular momentum is vector quantity and its SI unit is kg . $m^2$ rad/s

Linear Impulse

If force is constant, Impulse = F Δt
Impulse is a vector quantity and its SI unit is N.sec

Angular Impulse

If torque is constant, Angular Impulse = T Δt
It is a vector quantity and its SI unit is N.m.sec

Impulsive – Momentum Theorem for Particles

Impulse (M) = m ($v_2 – v_1$)

Application of Impulse – Momentum Theorem

$I_{1→2}$ = $M_2$ – $M_1$

Conservation of Momentum

∑m $v_2$ = ∑m $v_1$

Notes: The momentum is conserved,
1. when resultant of force is zero.
2. when time interval Δt is very small.
3. when all the external forces are non-impulsive.

Section II – Impact

Coefficient of Restitution (e)

$e = {v_2 – v_1}/{u_1 – u_2}$

Law of Conservation of Momentum

$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$

Perfectly Elastic Impact

e = 1
Both bodies separate after impact

Semi Elastic Impact

0 < e < 1
Both bodies separate after impact.

Plastic Impact

$m_1 u_1 + m_2 u_2 = v(m_1 + m_2)$

e = 0
Both bodies move together after impact.

Impact with infinite mass

$e = √{{h_2}/{h_1}}$
where,

$h_1$ = Height just before impact
$h_2$ = Height just after impact