Elastic collision in one dimension involves collision of two bodies, which move along the same straight line. In order that collision is one dimensional, it should be "head - on" collision so that bodies after collision follow the same straight line. This restriction is implied for one dimensional collision.

In this case, there are two governing equations with respect to conservation of linear momentum and kinetic energy. Since collision is in one dimension, we can use scalar form of linear momentum equation with a sign convention. The velocity in the reference direction of associated coordinate system is considered positive.

To write the governing equations, let us consider that

Elastic collision in one dimension |
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We can rearrange this equation by clubbing attributes of projectile and target on either side of the equation as :

From kinetic energy consideration, we have :

Again rearranging attributes of bodies on either side of equation :

Dividing equation - 4 by equation - 2, we have :

This is an important equation. This equation can be interpreted in terms of relative velocity with which projectile and target move with respect to each other. The left hand side of the expression gives the velocity of approach i.e. the relative velocity with which projectile moves towards the target for the collision to occur. On the other hand, right hand side of the equation gives the velocity of separation with which target is separated from the projectile.

We can summarize the discussion so far about elastic collision in one dimensions with following characterizing aspects :

- The bodies move along same straight line before and after collision.
- The collision is "head - on".
- There is no loss of kinetic energy of the system during collision.
- Velocity of approach (of projectile) is equal to velocity of separation (of target).

** Velocities after collision **

We can use following two equations to find expression for final velocities of two bodies after collision :

To find final velocity of projectile,

Subtracting equation - 6 from equation - 1, we have :

Similarly, we can find the final velocity of the target by multiplying equation - 5 by

** Special cases **

Here, we discuss some special cases of elastic collision in one dimension :

1: Collision between equal masses :

In this case,

and

This is an interesting result. When bodies of two equal masses collide elastically and "head - on", the velocities of projectile and target are exchanged. The projectile acquires the initial velocity of target and the target acquires the initial velocity of the projectile. If the target is initially at rest, then projectile is rendered to come to a dead stop as it acquires the initial velocity of the target. We can probably experience such thing if we are able to hit an identical billiard ball or coin "head - on" with identical billiard ball or coin. If the hit is "head - on", then the striking ball or coin should come to a stop.

2: Collision between light projectile and heavy target :

From our daily experience, we can visualize such collision. What happens when a hard ball (light projectile) hits a wall or hard surface (heavy target). We know that wall remains where it stands, whereas the ball reverses its trajectory without any change in the magnitude of velocity. Let us see whether this common experience is supported by the analysis. Here,

Putting these approximate values in the equations of final velocities (equations 7 and 8), we have :

and

The second result reveals that the heavier target maintains its initial velocity. If the target is initially at rest , then it remains at rest.

On the other hand, velocity of the lighter projectile varies and depends on the combination of initial velocities of both projectile and target. If the target is initially at rest, then

This means that the projectile reverses its motion.

3: Collision between heavy projectile and light target :

Could we think of such collision and the consequent result? It is likely that the motion of heavy projectile is not affected at all, whereas it is likely that the lighter target flies off with a higher velocity. It is what our common sense indicates. Let us test our common sense.

Here,

and

Putting these approximate values in the equations of final velocities (equations 7 and 8), we have :

and

The first result reveals that the heavier projectile indeed maintains its initial velocity.

On the other hand, velocity of the lighter target varies and depends on the combination of initial velocities of both projectile and target. If the target is initially at rest, then

This means that the light target flies off with double the initial velocity of the projectile.

We can summarize these results for elastic collision in one dimension as :

- Velocities are exchanged when bodies have equal masses.
- If the heavier target is at rest, then the motion of target remains same, whereas projectile reverses its direction without any change in magnitude.
- If the projectile is heavier and target at rest, the motion of projectile remains same, whereas target acquires double the initial velocity of the projectile.

We can generally conclude from above results that velocity of the heavier (projectile or target) is unaltered by collision. Most of the time, it is not possible to memorize the expressions of final velocities for different cases. It is generally more convenient to apply the governing laws for the given situation and solve the equation independently. However, the approximated results for the special cases as enumerated above helps save time for working with problems of elastic collision in one dimension. It is, therefore, convenient to remember the results of special cases as enumerated above.

##### Example 1

Problem : Two spheres "A" and "B" as shown in the figure are identical in shape, size and mass. The sphere "A" moves right with a velocity "v", whereas sphere "B" is at rest. If the wall on the right is fixed, find the velocities of the spheres after all possible collisions have taken place. Assume that collisions are elastic collisions in one dimension and no friction is involved between surfaces.

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Solution : Since collisions take place in one dimension, it is imperative that all collisions are "head on" collisions. In order to know the final states of the motion, we proceed from the first collision till the possibility of collision exhausts.

The sphere "A" collides "head - on" with sphere "B". It is one dimensional elastic collision involving equal masses. Therefore, the velocities of the colliding spheres are exchanged after collision. It means that sphere "A" acquires the velocity of "B" i.e it comes to rest. On the other hand, sphere "B" acquires the velocity of sphere "A" and moves with velocity "v" towards the fixed wall. The situation is shown in the figure below.

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The sphere "B" moving with velocity "v" collides with fixed wall. This is elastic collision of a lighter projectile with a heavy target. As such, the velocity of the colliding sphere "B" is simply reversed after collision. It means that sphere "B" moves towards sphere "A" with velocity "v" after reversing its motion subsequent to collision with fixed wall. This situation is depicted in the figure.

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The sphere "B" collides with stationary sphere "A". Again the collision being one dimensional elastic collision between spheres of equal masses, velocities are exchanged. The sphere “B” come to a stop, whereas sphere "A" starts moving left with velocity "v". This constitutes the last collision as sphere "A" moves away from the "B". The final situation is shown in the figure. Evidently, final situation is same as initial situation with only difference that sphere is moving towards left as against its motion towards left in the beginning.

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This example illustrates how we can analyze elastic collision situations in one dimension with the help of results arrived for special cases. Notably, solution completely avoids any mathematical analysis.