3.1aMass, weight and momentum
Definitions
- Mass
- the property of an object that resists change in its motion (its inertia).
- Weight
- the gravitational force acting on an object; W = mg.
- Linear momentum
- the product of mass and velocity.
- Newton
- the force that gives a mass of 1 kg an acceleration of 1 m s-2.
- Impulse
- the product of force and the time for which it acts; equal to the change in momentum.
Key points
- Mass is a scalar (kg), the same everywhere; weight is a vector force (N) that depends on g.
- Momentum is a vector in the direction of the velocity; sign matters when directions reverse.
- Unit of momentum: kg m s-1 = N s.
- Force-time graph: area under the graph = impulse = Δp.
- Same Δp over a longer time means a smaller force (crumple zones, bending knees, air bags).
Force-time graph for an impact: the shaded area is the impulse, equal to the change in momentum.
Watch out!
- Quoting weight in kg. Weight is a force: newtons.
- A ball bouncing back: Δp = m(v - u) with u and v of opposite sign, so the magnitudes add.
3.1bNewton's laws of motion
Definitions
- Newton's first law
- a body remains at rest or continues at constant velocity unless acted on by a resultant force.
- Newton's second law
- the resultant force on a body is proportional to (equal to) its rate of change of momentum, in the direction of the force.
- Newton's third law
- if body A exerts a force on body B, then B exerts a force on A that is equal in magnitude, opposite in direction and of the same type.
Key points
- Resultant force zero: no acceleration (at rest or constant velocity).
- Resultant force and acceleration are always in the same direction.
- For constant mass the second law gives F = ma; use the resultant force.
- Draw a free-body diagram for one object before writing F = ma.
- Third-law pairs act on different bodies, so they never cancel.
- Weight of a book and the table push on it are not a third-law pair (same body, different types).
Free-body diagram of an accelerating car: only forces ON the car; resultant = driving force - drag.
A third-law pair: equal, opposite, same type (gravitational), acting on different bodies.
Watch out!
- Using a single force (not the resultant) in F = ma.
- Saying a third-law pair "balances". Balanced forces act on the same body; third-law pairs never do.
- Constant velocity does not need a forward resultant force: resultant is zero.
3.2Non-uniform motion: drag and terminal velocity
Definitions
- Drag
- a resistive force on an object moving through a fluid, acting opposite to the velocity.
- Terminal velocity
- the constant (maximum) velocity reached when the drag force equals the driving force (weight), so the resultant force is zero.
Key points
- Drag and viscous forces increase with speed; they oppose the motion.
- At release: drag = 0, so a = g.
- As v rises: drag rises, resultant force falls, acceleration falls (v-t gradient decreases).
- Terminal velocity: drag = weight, resultant = 0, a = 0, v constant.
- Opening a parachute: drag > weight, so the body decelerates to a new, lower terminal velocity.
- Larger area or less streamlined shape: larger drag, lower terminal velocity.
Falling with air resistance: gradient g at release, flattening to the terminal velocity.
Falling body: at release only W acts; drag grows with speed until D = W (terminal velocity).
Watch out!
- At terminal velocity the acceleration is zero, not the velocity.
- Opening a parachute does not make the diver go up: the velocity stays downward while it decreases.
3.3Linear momentum and its conservation
Definitions
- Principle of conservation of momentum
- for a system of interacting bodies, the total momentum remains constant provided no resultant external force acts.
- Elastic collision
- a collision in which total kinetic energy is conserved (as well as momentum).
- Inelastic collision
- a collision in which total kinetic energy is not conserved (momentum still is).
- Perfectly inelastic collision
- an inelastic collision in which the bodies stick together and move with a common velocity.
Key points
- Momentum is conserved in all collisions and explosions (isolated system).
- Total energy is always conserved; in an inelastic collision KE becomes heat, sound, deformation.
- Elastic: relative speed of approach = relative speed of separation.
- Explosion from rest: total momentum stays zero, so fragments move in opposite directions.
- Two dimensions: conserve momentum separately in two perpendicular directions.
- Choose a positive direction; velocities the other way are negative.
Elastic collision check: p 6 = 6 kg m s-1, Ek 9 = 9 J, approach 3 = separation 3 m s-1.
Oblique collision: the momentum vectors after add tip-to-tail to give the momentum before.
Watch out!
- Testing momentum conservation with kinetic energy. KE is conserved only in elastic collisions.
- Forgetting the minus sign for a body moving in the negative direction after the collision.
- Relative speed rule applies to elastic collisions only.