Momentum and impulse in one and two dimensions, the impulse-momentum principle, conservation of momentum, and impulse as the area under a force-time graph.

What this dot point is asking

Edexcel Further Mechanics wants you to use momentum and impulse in one and two dimensions, apply the impulse-momentum principle, conserve momentum in collisions and explosions, and interpret impulse as the area under a force-time graph or the integral of force over time. Vector treatment in two dimensions and the link to Newton's second law are both examined.

Momentum and the impulse-momentum principle

Momentum $\mathbf{p} = m\mathbf{v}$ is a vector, so direction matters and signs (or components) must be tracked. The impulse-momentum principle is Newton's second law integrated over time: a force acting for a time produces a change in momentum equal to the impulse. For a constant force this is simply $\mathbf{F}t$; for a varying force it is the time integral, which on a force-time graph is the area under the curve.

Conservation of momentum

When no external impulse acts on a system (the collision or explosion forces are internal and cancel in pairs by Newton's third law), the total momentum of the system is unchanged. This single principle solves collisions, coalescences, explosions and recoil problems. Kinetic energy, by contrast, is generally not conserved in a collision.

Two dimensions

In two-dimensional problems momentum is conserved independently in each of two perpendicular directions. Resolve every velocity into components (often along and perpendicular to a line of impact, or along the $x$- and $y$-axes) and write a separate conservation equation for each direction. Vector impulses add component-wise to vector momenta.

Examples in context

Momentum and impulse are the foundation for the elastic-collisions dot point, where conservation of momentum is paired with Newton's law of restitution to solve impacts. The integral form $\mathbf{J} = \int \mathbf{F}\,dt$ links impulse to the calculus of further calculus and to force-time modelling. In rocket and recoil problems the same conservation law explains why a gun recoils when fired or why a stationary system splits into oppositely moving pieces. The connection to Newton's second law ($\mathbf{F} = \frac{d\mathbf{p}}{dt}$) ties impulse to differential equations of motion.

Try this

Q1. A force of $5\,\text{N}$ acts for $4\,\text{s}$. Find the impulse. [1 mark]

• Cue. $J = Ft = 5 \times 4 = 20\,\text{N s}$.

Q2. A $0.5\,\text{kg}$ ball changes velocity from $4$ to $-2\,\text{m s}^{-1}$. Find the impulse on it. [2 marks]

• Cue. $J = m(v - u) = 0.5(-2 - 4) = -3\,\text{N s}$.

Q3. A $4\,\text{kg}$ trolley at $2\,\text{m s}^{-1}$ collides and couples with a stationary $1\,\text{kg}$ trolley. Find the common speed. [2 marks]

• Cue. $4(2) = 5v$, so $v = 1.6\,\text{m s}^{-1}$.