Question
Two bodies undergo one$-$dimensional, inelastic, head$-$on collision. Obtain an expression for the magnitude of impulse.
✓
Answer
$i.$ When two bodies undergo collision, the linear momentum delivered by the first body to the second body must be equal to the change in momentum or impulse of the second body and vice versa.
$\therefore$ Impulse,
$|J|=\left|\Delta p_1\right|=\left|\Delta p_2\right|$
$=\left|m_1 v_1-m_1 u_1\right|=\left|m_2 v_2-m_2 u_2\right| .....(1)$
$ii.$ Substituting $v_1=\frac{m_1-e_2}{m_1+m_2} u_1+\frac{(1+e) m_2}{m_1+m_2} u_2$,
$v_2=\frac{m_2-e_1}{m_1+m_2} u_2+\frac{(1+e) m_1}{m_1+m_2} u_1$
In equation $(1)$ and solving, we get,
$|J|=\left(\frac{m_1 m_2}{m_1+m_2}\right)(1+e)\left(\left|u_1-u_2\right|\right)=\mu(1+e) u_{\text {relative }}$
$u _{\text {relative }}=\left| u _1- u _2\right|=$ velocity of approach.
$\therefore$ Impulse,
$|J|=\left|\Delta p_1\right|=\left|\Delta p_2\right|$
$=\left|m_1 v_1-m_1 u_1\right|=\left|m_2 v_2-m_2 u_2\right| .....(1)$
$ii.$ Substituting $v_1=\frac{m_1-e_2}{m_1+m_2} u_1+\frac{(1+e) m_2}{m_1+m_2} u_2$,
$v_2=\frac{m_2-e_1}{m_1+m_2} u_2+\frac{(1+e) m_1}{m_1+m_2} u_1$
In equation $(1)$ and solving, we get,
$|J|=\left(\frac{m_1 m_2}{m_1+m_2}\right)(1+e)\left(\left|u_1-u_2\right|\right)=\mu(1+e) u_{\text {relative }}$
$u _{\text {relative }}=\left| u _1- u _2\right|=$ velocity of approach.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Prove that: Vector addition is associative.
Discuss the effect of external voltage on the width of depletion region of a p-n junction.
Under what conditions is total internal reflection possible? Explain it with a suitable example. Define critical angle of incidence and obtain an expression for it.
Two weights 5 kg and 8 kg are suspended from a uniform rod of length 10 m and weighing 3 kg. The distances of the weights from one end are 2 m and 7 m. Find the point at which the rod balances.
A letter $‘E \ ’$ is prepared from a uniform cardboard with shape and dimensions as shown in the figure. Locate its centre of mass.
→What are electric lines of force?
How are X-rays produced? State their properties and uses.
Describe how an artificial satellite using two stage rocket is launched in an orbit around the Earth.
Explain the concept of additive nature of charge.
Explain how solids are classified on the basis of band theory of solids.