Dynamics

Syllabus points

• State and apply Newton's three laws of motion.
• Define linear momentum and relate resultant force to rate of change of momentum.
• Describe and use the concept of weight as the effect of a gravitational field on a mass.
• State the principle of conservation of momentum and apply it to collisions in one dimension.
• Distinguish between elastic and inelastic collisions.

Newton's laws

First law. A body remains at rest or moves at constant velocity unless acted on by a resultant force. So constant velocity and rest are the same state as far as forces go: both mean zero resultant force. A question saying a body moves at constant speed in a straight line is telling you the forces balance.

Second law. The resultant force is proportional to the rate of change of momentum and acts in the same direction:

F = Δp / Δt

For constant mass this becomes the familiar F = ma. The momentum form is the general one and is the one to quote when mass changes, as it does for a rocket or for a jet of water hitting a wall.

Third law. If body A exerts a force on body B, then B exerts an equal and opposite force on A.

The third law is the one most often stated wrongly. The two forces:

• act on different bodies, so they never cancel each other out
• are of the same type, so a gravitational pull is paired with a gravitational pull, never with a contact force
• are equal in magnitude and opposite in direction, always, whatever the masses

A book resting on a table is the classic trap. The weight of the book and the normal contact force from the table are not a third-law pair: they act on the same body and are different types. The pair for the book's weight is the gravitational pull of the book on the Earth. The pair for the contact force is the push of the book down on the table.

Weight

Weight is the force of gravity on a mass:

W = mg

Weight is a force measured in newtons and is a vector; mass is a scalar measured in kilograms and does not change with location. An astronaut's mass on the Moon is the same as on Earth; the weight is about a sixth.

g has two readings that are numerically equal: it is both the acceleration of free fall, 9.81 m s⁻², and the gravitational field strength, 9.81 N kg⁻¹.

Momentum

Momentum p = mv. It is a vector, measured in kg m s⁻¹ or equivalently N s, and its direction is the direction of the velocity.

Because it is a vector, direction must be handled with signs in every calculation. Choose a positive direction, write every velocity with a sign, and keep it.

Impulse

Impulse = FΔt = Δp

So the area under a force-time graph is the change in momentum. This explains crumple zones and airbags: they lengthen Δt for the same Δp, so the force is smaller.

Conservation of momentum

In the absence of external forces, the total momentum of a system is constant.

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

This follows from the third law: the forces the two bodies exert on each other are equal and opposite and act for the same time, so the impulses are equal and opposite, so the momentum gained by one is the momentum lost by the other.

Worked example. A trolley of mass 2.0 kg moving at 3.0 m s⁻¹ collides with a stationary trolley of mass 4.0 kg. They stick together. What is their common velocity?

Before: 2.0 × 3.0 + 4.0 × 0 = 6.0 kg m s⁻¹.

After: (2.0 + 4.0)v = 6.0v.

So v = 1.0 m s⁻¹ in the original direction.

Worked example with a sign change. A ball of mass 0.20 kg hits a wall at 6.0 m s⁻¹ and rebounds at 4.0 m s⁻¹. What is the magnitude of the change in momentum?

Taking the initial direction as positive, the initial momentum is +1.2 kg m s⁻¹ and the final is −0.8 kg m s⁻¹.

Δp = −0.8 − (+1.2) = −2.0 kg m s⁻¹, so the magnitude is 2.0 kg m s⁻¹.

Answering 0.4 by subtracting the magnitudes is the standard error, and it is worth noting that the change is larger than either momentum on its own. That is why a ball that bounces exerts a bigger force than one that stops dead.

Elastic and inelastic

Momentum is conserved in every collision. Only kinetic energy distinguishes the two, and the lost kinetic energy becomes internal energy and sound.

Perfectly elastic collisions are rare in the everyday world; they are the model for gas molecules. Any collision where bodies stick together is inelastic, because they cannot separate and so cannot recover the energy.

For a perfectly elastic collision there is a useful shortcut: the relative speed of approach equals the relative speed of separation. It gives a second equation to sit beside conservation of momentum, and it is quicker than writing out the kinetic energy equation with its squares.

Non-uniform motion and drag

A body falling through air experiences drag, which increases with speed. The resultant force is W − F_drag, so the acceleration falls as the speed rises, and the body approaches terminal velocity asymptotically.

The same reasoning applies to a car reaching top speed: the driving force is constant and the drag rises with speed until they balance.

Common mistakes

• Naming the weight of a book and the table's normal contact force as a third-law pair.
• Saying the third-law forces cancel out. They act on different bodies.
• Adding magnitudes instead of using signed velocities when a body rebounds.
• Saying momentum is not conserved in an inelastic collision. It always is.
• Treating kinetic energy as conserved when two bodies stick together.
• Confusing mass and weight, particularly in a question about a different planet.
• Using F = ma when the mass is changing; the momentum form is needed.