Newton's Laws, Momentum, Impulse and Friction

What you need to know

• Newton's first law states that a body remains at rest or continues in uniform motion in a straight line unless acted upon by an external force. This is the law of inertia, and it explains why a passenger lurches forward when a bus brakes suddenly.
• Inertia is the reluctance of a body to change its state of rest or motion, and mass is the measure of inertia. A loaded lorry is harder to start and harder to stop than an empty one for exactly this reason.
• Newton's second law states that the rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction of the force. In symbols F = (mv - mu)/t, which reduces to F = ma when mass is constant.
• Newton's third law states that to every action there is an equal and opposite reaction. The two forces are equal in magnitude, opposite in direction, and crucially they act on two different bodies, which is why they never cancel out.
• Momentum is mass times velocity, a vector measured in kg m/s or N s. Direction matters: taking rightward as positive, a body moving left has negative momentum, and ignoring this wrecks collision questions.
• The principle of conservation of linear momentum states that in a closed system with no external force, total momentum before collision equals total momentum after. Write m1u1 + m2u2 = m1v1 + m2v2 every time.
• In a perfectly inelastic collision the bodies stick together and move with a common velocity, so m1u1 + m2u2 = (m1 + m2)v. Momentum is conserved but kinetic energy is not; the difference is lost as heat, sound and deformation.
• In an elastic collision both momentum and kinetic energy are conserved, and the bodies separate. In an explosion the total momentum before is zero, so the fragments fly apart with equal and opposite momenta, which is how recoil of a gun is calculated.
• Impulse is force times the time for which it acts, and impulse equals the change in momentum: Ft = mv - mu. This is why a boxer rides a punch and a long jumper lands in sand: increasing the stopping time reduces the force for the same change in momentum.
• Friction is the force that opposes relative motion between two surfaces in contact. Static friction acts before motion begins and rises to a maximum called limiting friction; kinetic (dynamic) friction acts once sliding has started and is slightly smaller.
• Limiting friction F = mu N, where mu is the coefficient of friction and N is the normal reaction. On a horizontal surface with no vertical pull, N = mg, so F = mu m g. Friction depends on the nature of the surfaces and on N, but not on the area of contact or on the speed.
• On a plane inclined at angle theta, the component of weight down the plane is mg sin theta and the normal reaction is mg cos theta. A body on the point of sliding gives mu = tan theta, where theta is the angle of friction or angle of repose.
• Friction is useful in walking, braking, gripping the road, belt drives and writing; it is wasteful in machine bearings and engines, where it is reduced by lubrication, ball bearings, streamlining and smoothing the surfaces.
• Weight is the force of gravity on a body, W = mg, measured in newtons and varying with location. Mass is the quantity of matter in a body, measured in kilograms and constant everywhere. A 60 kg student weighs 600 N on earth using g = 10 m/s^2.