Linear momentum

An object’s linear momentum depends on its mass and velocity. It is calculated by multiplying the mass by the velocity:

Here, $$p$$ is momentum, $$m$$ is mass and $$v$$ is velocity. The unit of momentum is $$\mathrm{kg,m,s^{-1}}$$.

Momentum is a vector: it has both magnitude and direction. Because mass is positive, momentum points in the same direction as velocity.

Momentum in one dimension

For motion along a straight line, choose one direction as positive. Velocity and momentum are positive in that direction and negative in the opposite direction.

For example, if right is positive, an object of mass $$2,\mathrm{kg}$$ moving left at $$3,\mathrm{m,s^{-1}}$$ has velocity $$-3,\mathrm{m,s^{-1}}$$ and momentum $$-6,\mathrm{kg,m,s^{-1}}$$. The negative sign shows that its momentum points left.

Conservation of linear momentum

The total linear momentum of a system stays constant when the resultant external force on the system is zero. A system is the collection of objects being considered; an external force is a force exerted on it by something outside that collection.

Objects within the system can exert forces on one another and exchange momentum. Their individual momenta may therefore change, even though their total momentum remains constant.

An object's momentum is $$p = mv$$, where $$m$$ is its mass and $$v$$ is its velocity. Momentum has the same direction as velocity.

Momentum in one dimension

In a one-dimensional collision or explosion, choose one direction as positive and use it consistently. An object moving in the opposite direction has negative velocity and therefore negative momentum. Add the momenta with their signs: if the resultant external force is zero, the signed totals satisfy

For example, suppose a $$2,\mathrm{kg}$$ object moving at $$3,\mathrm{m,s^{-1}}$$ collides with a stationary $$1,\mathrm{kg}$$ object. If they stick together and the resultant external force is zero, their common velocity $$v$$ is found from

Taking the first object's direction as positive gives $$v = 2,\mathrm{m,s^{-1}}$$. The total momentum is $$6,\mathrm{kg,m,s^{-1}}$$ both before and after the collision.

An initially stationary explosion

If a system is initially stationary, its total momentum is zero. If it explodes while the resultant external force on it is zero, its total momentum afterwards must also be zero.

The parts can nevertheless move with non-zero momenta. For two parts, their momenta must be equal in magnitude and opposite in direction: $$p_1 = -p_2$$. Their signed momenta add to zero; their velocities do not have to be equal in magnitude if their masses differ.

Elastic and inelastic collisions

When objects collide, their velocities can change. To classify the collision, compare the total kinetic energy of all the colliding objects immediately before and immediately after the collision.

In an elastic collision, total kinetic energy is conserved: the total after the collision is equal to the total before it. The kinetic energy of an individual object may still change; it is the combined total that stays the same.

In an inelastic collision, total kinetic energy is not conserved. Some kinetic energy is transferred to other forms, such as internal energy associated with deformation or heating. Total energy is still conserved, but not all of it remains as kinetic energy.

Momentum in collisions

Total momentum is conserved in both elastic and inelastic collisions when the resultant external force on the colliding system is zero. The objects can transfer momentum to one another, but their combined momentum remains the same.

Momentum conservation therefore does not mean that kinetic energy is also conserved. The distinction between elastic and inelastic collisions is whether the system’s total kinetic energy stays the same.

Impact forces and contact time

An impact changes an object’s momentum while a force acts on it. The force acts over a period called the contact time. For example, a football’s momentum changes while it is in contact with a player’s foot.

The average force depends on both the change in momentum and the time taken for that change:

Here, $$F_{\mathrm{avg}}$$ is the average force, $$\Delta p$$ is the change in momentum, and $$\Delta t$$ is the time over which it occurs. For the same change in momentum, increasing the contact time reduces the magnitude of the average force. For example, doubling the time halves the average force.

Extending impact time

When a player kicks a football, the foot exerts a force on the ball during contact, changing its momentum. If the same momentum change occurs over a longer contact time, the average force is smaller.

A crumple zone deforms during a collision, extending the time over which the vehicle and its occupants slow down. For a given change in momentum, this reduces the average force associated with slowing them. Similarly, protective packaging compresses during an impact, extending the time over which the packaged object slows and reducing the average force on it.

A smaller average force does not mean that momentum has been eliminated: the momentum has changed through forces acting during the collision. Nor does a reduced average force guarantee that injury or damage will be avoided.

Ethical transport design

Designers must consider the foreseeable effects of collision protection on the people affected, including occupants and other road users. Extending the time over which momentum changes can reduce average impact forces, but the consequences for everyone involved must be considered.