momentum / impulse / conservation of momentum
Momentum
Momentum is defined as the mass of an object multiplied by the velocity.
p= mv
Since the momentum depends on the velocity, it is also a vector quantity. The units for momentum are kg.m.s^{-1} or Ns.
Momentum can also be described in terms of Newon's Second Law. If we measure the momentum of an object over a small period of time.
Since (v - u) / t = acceleration
F = ma
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Impulse
Taking the time to the left-hand-side of the equation above,
Ft = mv - mu
The force multiplied by the time has a special name, the **impulse**. The impulse is the change in the momentum. Impulse is a vector with units of kg.m.s^{-1} or Ns.
Impulse is the force multiplied by time the force acts on an object.
Impulse = Ft = Δp
Consider a ball that is thrown into the air. We can make the ball go higher in two ways, firstly, we can apply a larger force, F or we can make the same force act on the ball for a longer time.
If the force acting on the object is not constant over the time for the time in which the force acts, the impulse is given by the integral of the force with respect to time.
In collisions between moving objects, the magnitude of the impulse is what changes each object's velocity.
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Conservation of Momentum
Conservation is an important principle in physics in that it allows us to determine what happens in collisions or explosions. If a body A of mass m_{A} and velocity u _{A} collides with another body B of mass m_{B} and velocity u _{B} moving in the same direction. If A exerts a force F to the right on B for a time t then by Newton's third law, body B will exert an equal and opposite force to the left on A also for a time t but to the left. Thus the bodies receive equal but opposite impulse Ft. The changes of momentum must be equal and opposite. Therefore, the total momentum change is zero. In other words, the total momentum of A and B together remains constant. In mathematical terms:
m_{A}u_{A}+ m_{B}u_{B} = m_{A}v_{A}+ m_{B}v_{B}
This is the general case. For motion in 1-dimension, we can consider the momenta in one direction only which allows us to drop the vector notation and use positive and neagtive signs.
It is important to release that the momentum is conserved only when there are no external agent acting on the system that could add momentum to the system. They must be included in the system before the conservation of momentum can be applied.
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download worksheet 8.4.4.A - momentum
download worksheet 8.4.4.B - collisions
download experiment 7 - consveration of momentum |