What is momentum?

A light trolley moving quickly and a heavier trolley moving slowly can have the same momentum. Momentum combines mass with velocity and describes motion as a vector quantity: both its magnitude and direction matter. For ordinary mechanical speeds, the relationship is p = mv.

For instance, a 2 kg trolley travelling right at 3 m/s and a 4 kg trolley travelling right at 1.5 m/s each have a momentum of +6 kg·m/s. They do not have the same kinetic energy, and they will not necessarily stop over the same distance under the same braking force.

Calculating momentum with consistent units

To calculate momentum, use mass in kilograms and velocity in metres per second. The SI unit of momentum is kg·m/s, equivalent to N·s. Mass is not weight: weight is a force. If a speed is supplied in km/h, divide it by 3.6 before using the formula.

Choose a positive direction before adding momenta. A negative answer means motion in the opposite direction; it does not indicate a negative mass or an impossible result. The examples and tool below use rightward motion as positive.

Zero total momentum can coexist with motion

The tool starts with A moving right and B moving left. Their momenta are +6 and −6 kg·m/s, so the vector sum is zero. Both objects are moving and both have kinetic energy. Their momentum magnitudes are equal, but their individual momentum vectors point in opposite directions. In everyday momentum examples, identifying the objects included in the system is just as important as calculating their values.

What determines the force needed to stop an object?

Stopping requires a change in momentum. The impulse relationship is J = Δp = F_avgΔt, where F_avg is the average net force along the direction being analysed. Contact forces can also produce momentum transfer between objects.

Suppose a 12 kg carriage slows from +0.75 m/s to rest. Its momentum changes by −9 kg·m/s. Completing that change in 0.15 s requires an average net force of −60 N; allowing 0.45 s reduces the required average to −20 N. These are average forces, not peak impact loads, and momentum alone does not specify the stopping distance.

When is momentum conserved?

The law of conservation of momentum applies to the total momentum of a defined system. It remains constant when the net external force is zero. If the external impulse over an event is negligible, the initial and final totals can be treated as approximately equal. Merely calling a system “closed” does not establish that force condition.

Internal interactions can redistribute momentum. Two objects may exchange momentum during contact while the total stays unchanged. Significant friction from the floor or a force from an external stop changes the momentum of the two-object system, so those effects must be considered.

Elastic collisions conserve kinetic energy too

The difference between elastic and inelastic collisions concerns kinetic energy. In an elastic collision it is conserved; in an inelastic collision some becomes internal energy, deformation, heat or sound. Total energy does not disappear. In either case, momentum conservation depends on the external impulse acting on the selected system.

If two objects stick together, the collision is perfectly inelastic. For a one-dimensional event with negligible external impulse, their common final velocity is v_final = (m_Av_A + m_Bv_B) / (m_A + m_B). With the tool's initial equal and opposite momenta, sticking together would leave the pair at rest. The displayed animation uses separate lanes and does not model this collision.

Momentum, energy and power are different quantities

The relationship between momentum and energy follows from p = mv and E_k = ½mv². Doubling velocity doubles momentum magnitude but quadruples kinetic energy. Momentum is a vector; kinetic energy is a scalar measured in joules.

The two 6 kg·m/s examples at the top have kinetic energies of 9 J and 4.5 J. Equal momentum magnitude therefore does not mean equal energy. An electric motor power rating is different again: power measures the rate of energy transfer and is expressed in watts.

Rotating machinery uses angular momentum

Linear and angular momentum describe different aspects of motion. A particle's angular momentum about a chosen origin is L = r × p. For a rigid body rotating about a specified axis, the component along that axis can be calculated as L = Iω, using the moment of inertia I about the same axis.

Angular momentum is measured in kg·m²/s. Convert a speed in RPM to angular velocity with ω = 2πn/60 before using it in this calculation. Net external torque determines the rate of change of angular momentum. Conservation of angular momentum requires zero net external torque about the relevant reference.

How this relates to motion control

A servo motor must accelerate its rotor and the connected load. A servo system therefore needs information about inertia and the requested motion time, as well as the target speed. A linear encoder can track linear motion and a suitable incremental rotary encoder can track rotation. Position feedback alone does not supply the mass or inertia needed for momentum calculations.

A drive's deceleration ramp affects the required braking torque. With the same inertia and speed change, a shorter stopping time requires greater average net braking torque. The drive's constant-torque operating region and the load torque both need consideration. The energy path also matters: in a suitable drive arrangement, a braking resistor can dissipate recovered mechanical energy as heat.