JEE Physics Collisions Notes Set 01

What Happens in a Collision

A collision is a short interaction in which two bodies exert large forces on each other and change their motion. The bodies need not actually touch; what matters is that the interaction time is very short, so the internal forces are huge compared with any external force. During the interaction the total linear momentum of the colliding bodies is conserved. Mechanical energy may or may not be.

Types of Collision

By direction of motion: a head-on (one-dimensional) collision happens along one straight line before and after impact; an oblique (two-dimensional) collision has the velocities at angles to the line of impact.
By energy: in an elastic collision total kinetic energy is unchanged; in an inelastic collision some kinetic energy is lost; in a perfectly inelastic collision the bodies move off together and the loss is the largest possible.

Coefficient of Restitution

Newton's experimental law compares the relative speed after impact with the relative speed before it, measured along the line of impact:
\[ e=\frac{v_2-v_1}{u_1-u_2}=\frac{\text{speed of separation}}{\text{speed of approach}} \]
e = 1 for an elastic collision, e = 0 for a perfectly inelastic one, and 0 < e < 1 otherwise. It depends on the materials of the bodies.

Head-on Collision Formulae

Using momentum conservation and the definition of e, the final velocities are
\[ v_1=\frac{m_1u_1+m_2u_2-m_2e(u_1-u_2)}{m_1+m_2},\qquad v_2=\frac{m_1u_1+m_2u_2+m_1e(u_1-u_2)}{m_1+m_2} \]
The kinetic energy lost is \( \Delta K=\frac{m_1m_2}{2(m_1+m_2)}(1-e^2)(u_1-u_2)^2 \).
Special cases for e = 1:
Equal masses: the bodies exchange velocities.
Heavy body hits a light body at rest (\( m_1\gg m_2,\ u_2=0 \)): \( v_1\approx u_1,\ v_2\approx2u_1 \).
Light body hits a heavy body at rest (\( m_2\gg m_1,\ u_2=0 \)): the light body rebounds with its own speed reversed and the heavy body stays put.
If e = 0 the bodies stick together: \( V=\frac{m_1u_1+m_2u_2}{m_1+m_2} \).

Oblique Collision

Resolve velocities along the line of impact (normal) and perpendicular to it (tangent). Along the normal, momentum is conserved and the restitution law applies. Along the tangent no impulse acts on either body, so each body's tangential velocity component is unchanged.
For a ball hitting a fixed smooth floor at angle \( \theta \) to the normal with speed v: the tangential component \( v\sin\theta \) stays the same and the normal component becomes \( ev\cos\theta \). Hence \( \tan\theta'=\frac{\tan\theta}{e} \), so the angle of reflection is greater than the angle of incidence when e < 1.

Bouncing Ball and Ballistic Pendulum

A ball dropped from height h onto the ground rebounds to \( e^2h \) after the first bounce and to \( e^{2n}h \) after the n-th. Hence \( e=\sqrt{h_2/h_1} \). The total distance travelled until it stops is \( h\frac{1+e^2}{1-e^2} \) and the total time is \( \sqrt{\frac{2h}{g}}\,\frac{1+e}{1-e} \).
In a ballistic pendulum a bullet embeds in a block (momentum conserved, energy lost), and then the block with the bullet swings up, converting kinetic energy into gravitational potential energy: \( \frac12(M+m)V^2=(M+m)gh \).

Spring Collisions

When two bodies interact through a spring, the spring is most compressed at the instant when both bodies have the same velocity (relative velocity zero). At that instant momentum gives the common velocity, and the loss of kinetic energy equals the energy stored in the spring: \( \frac12kx_{max}^2=\Delta K \). After the spring returns to its natural length, the collision is elastic.