Force and Momentum

Chapter 1

Reminders from GCSE

• Momentum is a measure of how easy or difficult it is to change the motion of a body– The greater the momentum, the bigger the

force needed to change it

• Momentum (p) = mass x velocitykg ms–1 or Nm kg ms–1

• Momentum is a vector

• Momentum is conserved

Newton’s Laws and momentum

• N1: An object remains at rest or travelling at a constant velocity unless acted on by a force– ie a force is needed to change a body’s

momentum

• N2: the rate of change of momentum is proportional to the force acting

t

mumv

t

uvmmaF

We define the Newton as the unit of force which gives a mass of 1 kg an acceleration of 1 ms–2

More on Newton’s 2nd law

• More generally:

– If m is constant:

– If m changes at a constant rate:• e.g., a rocket ejecting hot exhaust gases

t

mvF

mat

vmF

t

mvF

Impulse

impulse (Ns)

• So impulse is equal to the change of momentum of a body

• This idea is used a lot in road safety– Collisions often involve large changes of momentum– If you can extend the time over which this happens,

you can reduce the force (and so serious injuries)

)(

so ,

mvtFt

mvF

Road safety

• All the devices shown below are designed to increase the time of the momentum change during an accident. How?

Impulse example

• A golf ball of mass 0.05 kg is hit off a tee at a speed of 40 ms–1. What is its momentum?p = mv = 0.05 × 40 = 2 kg ms–1

• The club was in contact with the ball for0.5 ms. What force did it exert on the ball?∆p = force × time, F = ∆p/t = 2/0.0005 F = 4000 N

– Golf club animation

Duck and airliner

• Estimate the impact force of a duck hitting an airliner.– Mass of duck = 0.5kg– Length of duck = 0.3m– Velocity of airliner =

250ms-1

• Equivalent to ~10.6 tonnes! kNF

t

vmF

104250/3.0

2505.0

Force-time graphs

• Force x time = change in momentum

• So area under graph = impulse

Rebound impacts

+u

-v

t

muF

t

mumv

t

pF

mumvp

2 u, vif

Rebound impacts

• For rebounds at an angle, need to consider normal components of velocity

• If u=v, 1=2

• Before collision unormal=ucos

• after collision vnormal=-ucos

• So p=-2mucos• F=-2mucos/t

u

v

Conservation of momentum

• The principle states: for a system of interacting objects, the total momentum remains constant, provided no external force acts.

• Derived by Newton from N3, but in fact more fundamental.

Conservation of momentum

• Force F1 on ball A:

• Force on ball B:

• But F1=-F2, so

t

umvmF AAAA

1

uA uB

vA vBt

umvmF BBBB

2

BBAAAABB

AAAABBBB

umumvmvm

umvmumvm

or Total momentum after Total momentum before

A

A

A

B

B

B

Conservation of momentum

• Now make sure you can do the questions on p. 13… by doing them

• …and q.4 on p. 20 – do it too.

Newton’s Cradle

• Flash animation

• More than you ever wanted to know here

Elastic collisions

• An elastic collision is one where there is no loss of kinetic energy– If a ball bounces perfectly elastically, it will

reach the same initial height

• In (macroscopic) real life there are no perfectly elastic collisions– but some gas particles and sub-atomic

particles get pretty close

• So Elastic means p and KE are conserved– Newton’s cradle is a good example

Head-on elastic collisionsObjects bounce off each other

Inelastic collisions

• In an inelastic collision, some KE is converted to other forms of energy– Heat, sound, light etc…

• A totally inelastic collision is one where the colliding objects stick together– Loss of KE is a maximum (but generally not complete)

• A partially inelastic collision is where the colliding objects move apart and have less KE after the collision than before.

Inelastic collisions

• Check you can do the calculations on page 15

Elastic collisions

Inelastic collisions

Centre of mass

• In all closed systems, the motion of the centre of mass is unchanged during a collision

• In an elastic collision there is motion relative to the centre of mass afterwards

• In a completely inelastic collision there is no motion relative to the centre of mass afterwards

• Adjustable applet• Billiard balls animation• Physclips

Explosions

• Momentum is conserved (as usual)

• Momentum before = momentum after = 0

• Make sure you can do qs on p. 17…