Chapter 9 - Force and momentum (9.1 - Momentum and impulse (Momentum…
Chapter 9 - Force and momentum
9.1 - Momentum and impulse
Defined as its mass x its velocity
The unit of momentum is kgms.-2
Momentum is a vector quantity. Its direction is the same as the direction of the object's velocity.
Momentum, p = mv
Momentum and Newton's laws of motion
Newton's first law of motion: An object remains at rest or in uniform motion unless acted upon by a force
Newton's second law of motion: The rate of change of momentum of an object is proportional to the resultant force on it. In other words, the resultant force is proportional to the change of momentum per second.
F = change in momentum/change in time
If m is constant then F = change in mv, is m x change in v / change in t = ma
If m changes at a constant rate as a result of mass being transferred at a constant velocity then F = change in mv = v x change in m. Change in m/ change in time is change of mass per second.
The impulse of a force is defined as the force x the time for which the force acts.
Therefore, for a force F which acts for change in time,
The impulse = F x change in time = change in mv
Force - time graphs
F = mv - mu / t
rearranged to Ft = mv - mu
The area under the line of a force-time graph represents the change of momentum or the impulse of the force
9.2 - Impact forces
Force-time graphs for impacts
The variation of an impact force with time on a ball can be recorded using a force sensor connected using suitably long wires or a radio link to a computer. The force sensor is attached to the object that causes the impact. Because the force on the bat is equal and opposite to the force on the ball during the impact, the force on the ball due to the bat varies in exactly the same way as the force on the bat due to the ball. The variation of force with time is displayed on the computer screen.
When a ball hits a wall and rebounds, its momentum changes direction due to the impact. If the ball hits the wall normally, it rebounds normally so the direction of its momentum is reversed. The velocity and therefore the momentum after the impact is in the opposite direction to the velocity before the impact and therefore has the opposite sign.
Suppose the ball hits the wall normally with an initial speed u and it rebounds at speed v in the opposite direction. Since its direction of motion reverses on impact, a sign convention is necessary to represent the two directions. Its initial momentum is +mu and its final momentum is -mv.
F = change in momentum / change in time
F = -mv - mu / change in time = The impact force
9.3 - Conservation of momentum
Newton's third law of motion
When two objects interact, they exert equal and opposite forces on each other
The principle of conservation of momentum
The principle of conservation of momentum states that for a system of interacting objects, the total momentum remains constant, provided no external resultant force acts on the system.
the total final momentum = the total initial momentum
Testing conservation of momentum
This involves using two trolleys where the mass of each trolley is recorded beforehand. The first trolley is at rest and the second is pushed towards it at a constant velocity. Using a motion sensor a computer displays the velocity of the trolley throughout. After the impact the velocity of the trolleys dropped instantly. Using the measurements you can see that the initial momentum and the final momentum is the same.
(m.b + m.a)V = m.a x u.a
9.4 - Elastic and inelastic collisions
An elastic collision is one where there is no loss of kinetic energy
An inelastic collision occurs where the colliding objects have less kinetic energy after the collision than before the collision
9.5 - Explosions
When two objects fly apart after being initially at rest, they recoil from each other with equal and opposite amounts of momentum. So they move away from each other in opposite directions