Analysis of Collision

Changes during impact

(a) During impact each of the colliding bodies experiences a strong force.

Therefore, during the period of impact each of the bodies changes its momentum, but as a whole, the total momentum of the system does not change.

(b) Impact is practically followed by emission of light, sound, heat etc.

Therefore, during an impact or collision, the mechanical energy of the system does not remain constant whereas the total energy of the system remains unchanged.

(c) In ideal collisions such as collision between gas molecules, atoms, electrons, protons etc., the K.E. of the system of colliding particles remains constant before and after the impact. This type of collision is known as perfectly elastic collision. Remember that, during collision, the K.E. of the particles changes due to large impulsive forces.

(d) In some collisions, the K.E. of the system changes. In this case we cannot term the collision perfectly elastic.

(e) For perfectly elastic collisions, the K.E. of the system before and after the impact does not change, but during the impact, some of the K.E. is converted to elastic energy.

(f) If the colliding particles/bodies stick together, they move together with same velocity. This is a perfectly inelastic collision.

In this process sometimes K.E. is lost completely, for example, in dropping a stone into mud, the stone loses its total K.E. Sometimes, a fraction of the K.E. is lost,

for example, in the collision of a bullet and a hanging target.

(g) Relative velocity of the bodies just after the collision (velocity of separation) may or may not equal that before the collision along the line of impact (velocity of approach).

Illustration : A ball of mass m strikes the fixed inclined plane after falling through a height h,. If it rebounds without losing energy, find the impulse on the ball.

Solution:

N’N is perpendicular to the inclined face. N’N is therefore the line of impact.PQ-> is direction of initial velocity of the ball.

QR-> is the direction of final velocity of the ball.

Because this is an elastic collision,

∠PQN’ = ∠N’QR = θ

Impulse in measured along the impact line.

The change in velocity of the ball along NN’ = Δv = 2vCosθ (as explained earlier) where as there is no change in its velocity along the inclined plane.

⇒ Impulse = mΔv

⇒ Impulse = m(2v cosθ)

⇒ Impulse = 2mv cosθ

Since the ball falls through a height h just before collision

$ \displaystyle v = \sqrt{2 g h} $

⇒ Impulse = $ \displaystyle 2 m cos\theta \sqrt{2 g h} $

Coefficient of restitution

Newton proved by experiments that, when two bodies collide, the relative velocity after the impact bears a constant ratio with the relative velocity before impact along the line of action of impact. This constant is known as coefficient of restitution of the impact, denoted by the letter ‘ e ‘

$ \displaystyle e = \frac{velocity \, of \, separation(after\, collision)}{velocity\, of\, approach (before \, collision)} $

$ \displaystyle e = \frac{\vec{v_2}-\vec{v_1}}{\vec{u_1}-\vec{u_2}} $

(a) for perfectly elastic e = 1

(b) For perfectly inelastic v2 = v1 because the bodies move combiningly

v2 − v1 = 0

⇒ e = 0

(c) for other collision

0 < e < 1

(d) In general

0 ≤ e ≤ 1 (theoretically)

Note: Neither perfectly elastic nor perfectly inelastic collisions occur in nature. e is a dimensionless quantity.

Exericse : A particle of mass m moves with vo = 20 m/sec. towards a wall that is moving with v = 5 m/sec.

If the particle collides with the wall without losing its energy, find the speed of the particle just after the collision.

Also Read :

Impulse & Impulsive Force
Conservation of linear momentum during impact
Solved Examples : Impulse
Analysis of Collision & Coefficient of restitution
Line of impact during collision
Head on Collision , Velocities after Collision
oblique impact in Collision

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