# Momentum and Collisions

This article is a topic within the subject Higher Physics 1A.

## Contents |

## Momentum

^{[1]}**Momentum** is the product of an object's mass and velocity.

**p = mv**

Momentum is a vector, with the same direction as the velocity.

### Impulse

^{[2]}The **impulse** on a particle is equal to the **change** in momentum of a particle.

**Δp = I**

The **impulse** (I) is due to a net force (ΣF) acting on the particle. Impulse may be calculated by integrating the net force with respect to time:

**I = ʃΣFdt** (the limits of integration are the time over which the force acts)

If the force is constant, then the integration simplifies to:

**I = ΣFΔt** (Δt is the time over which the force acts).^{[3]})

### Conservation of Momentum

If no external forces act in a direction, momentum is **conserved** in that direction. If momentum is conserved, then it remains constant. Don't confuse this with conservation of mechanical energy, which is different (although mechanical energy and momentum may both be conserved in some situations^{[4]}).

## Centre of Mass

^{[5]}Most objects are not single particles, but rather are finite bodies with their mass distributed across the body, or systems of particles in a container. The motion of these objects may be described using their **centre of mass**. When a net force acts on an object, the object undergoes **translational motion** as though its entire mass is concentrated at its **centre of mass**. This is not affected other types of motion (such as rotation or vibration) or by deformation of the object.

**r _{cm} = Σ(m_{i}r_{i})/Σm_{i}**

The displacement of the centre of mass is given by "r_{cm}". To find this point, each individual mass (m_{i}) is multiplied by its displacement (r_{i}) and the sum of these values is taken. This is then divided by the total mass (or the sum of individual masses, Σm_{i}). Note that this equation will have to be applied in both x and y directions, to find both coordinates of the centre of mass.

For a continuous body, integration may be used:

**r _{cm} = (1/Σm_{i})ʃrdm**

## Multi-particle systems

^{[6]}A multi-particle system is one in which there are multiple particles (multiple masses). If a system has constant mass M, then it will respond to external forces as though it were a single particle of mass M, located at the system's centre of mass. The total momentum of the system (p_{tot})is equal to the sum of individual momenta (p_{i}) in the system:

**p _{tot} = Σp_{i}**

The total momentum may be described as the product of the system's mass (M) and the velocity of the centre of mass (v_{cm}):

**p _{tot} = Mv_{cm}**

Furthermore, all internal forces appear in pairs and cancel out (Newton's third law) so only external forces will affect the system. In other words, the change in momentum of the system (Δp_{tot}) is equal to the total impulse (I) imparted to the system by external forces:

**Δp _{tot} = I**

Note that if the impulse is zero in any direction, then no external forces act in that direction and so momentum is **conserved** in that direction. This means that any interactions between particles in the system will conserve momentum.

## Collisions

^{[7]}Collisions occur when two objects interact by means of forces. Most often this represents two object physically striking one another, but another example is the slingshot effect, using gravity. Any external forces are assumed to be negligible compared to the internal forces of the interaction, so momentum is **conserved**.

### Collisions in one dimension

^{[8]}Momentum is conserved during a collision. The momentum of the system before (i) and afterwards (f) will be the same. In equation form, for a collision between two objects (1 and 2):

**p _{1,i} + p_{2,i} = p_{1,f} + p_{2,f}**

**(m _{1}v_{1})_{i} + (m_{2}v_{2})_{i} = (m_{1}v_{1})_{f} + (m_{2}v_{2})_{f}**

These equations are useful for determining the motion of objects before and after collisions. Depending on whether an equation is **elastic** or not, other equations are produced which are also useful in determining colliding the objects' behaviour.

### Elastic and inelastic collisions

^{[9]} Whether a collision is **elastic** or **inelastic** indicates whether or not kinetic energy is **conserved** during the collision. The kinetic energy of the system's centre of mass is constant throughout the collision, but the kinetic energy of the individual particles *relative* to the centre of mass may change. This means the total kinetic energy of the system may or may not be conserved (note that momentum *is* conserved in collisions regardless of kinetic energy).

- In an
**elastic**collision, all kinetic energy is**conserved**(non conservative forces do no work).- The particles do not stick together at all in the collision.
- In reality, some kinetic energy is always lost as heat or sound, so no collision is truly elastic.
- The constant kinetic energy of elastic collisions may be represented using the equation:

- In an

**(½m _{1}v_{1}^{2})_{i} + (½m_{2}v_{2}^{2})_{i} = (½m_{1}v_{1}^{2})_{f} + (½m_{2}v_{2}^{2})_{f}** , since K = ½mv

^{2}

- In an
**inelastic**collision, kinetic energy is transferred to other forms (**not**conserved).- The objects will partly stick to one another.
- Most collisions are inelastic.
- A
**perfectly inelastic**collision is one where**all**kinetic energy is lost, so the objects will not separate after the collision. - A
**perfectly inelastic**collision may be described using the following equation (since all the mass is together after the collision):

- In an

**(m _{1}v_{1})_{i} + (m_{2}v_{2})_{i} = (m_{1} + m_{2})v_{f}**

### Collisions in two dimensions

^{[10]}Collisions occur in **two dimensions** when particles are no longer constrained to move in a single direction. One common example is a *glancing* collision, where one object strikes another off-center, so the objects both change direction *and* momentum.

**Two-dimensional** collisions are best dealt with by splitting the collision up into **x** and **y** components. Momentum is conserved in the **x** and **y** directions, and so the equations describing conservation of momentum may be used for each direction. The problem then becomes one of simultaneous equations.

**Example 1 - elastic collision in two dimensions**

In the x direction: m_{1}v_{1i}= m_{1}v_{1f}cosθ + m_{2}v_{2f}cosφ

In the y direction: 0 = m_{1}v_{1f}sinθ + m_{2}v_{2f}sinφ

**Example 2 - perfectly inelastic collision in two dimensions**

In the x direction: m_{1}v_{1i}= (m_{1}+ m_{2})v_{f}cosθ

In the y direction: m_{2}v_{2i}= (m_{1}+ m_{2})v_{f}sinθ

## End

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## References

**Textbook** refers to Serway & Jewett, *Physics for Scientists and Engineers* (Brooks/Cole , 8th ed, 2010)

**(Slides)** refers to those distributed by Wolfe, J (2012) on his First Year Physics site

- ↑ Textbook, pp235-236
- ↑ Textbook, pp240-241
- ↑ (Slides), Mechanics for systems of particles and extended bodies, p11
- ↑ (Slides), Mechanics for systems of particles and extended bodies, p12
- ↑ Textbook, pp253-255
- ↑ Textbook, pp258-259
- ↑ Textbook, pp242-243
- ↑ Textbook, pp242-243
- ↑ Textbook, pp243-244
- ↑ Textbook, pp250-251