Ch
Ch.7 Impulse and Momentum
Momentum is one of the most fundamental concepts in physics. It is a vector and is equal to mass times velocity. Or p = m.v. The momentum vector has the same direction as the velocity vector. Units of momentum are kg.m/sec. If mass is doubled, momentum also doubles; if velocity doubles momentum also doubles. If an object is at rest, it has zero momentum.
Law of conservation of momentum : states that when no net external force acts on a system (an isolated system), and no mass enters or leaves the system (closed) then the total linear momentum of the system is conserved. Momentum is thus called “inertia in motion”. For example, for an apple falling to the Earth, the velocity is constantly increasing and therefore the momentum is increasing, but the external force in this case is the gravitational force of the Earth. However, if we consider both the Earth and the apple as our system, there is no net external force. In such a case, the upward momentum of Earth is cancelled by the downward momentum of the apple, so the net total momentum is still conserved.
Impulse – Momentum theorem : states that an impulse produces a change in the object’s momentum. Impulse = Force x time or, I = F.t. Change in momentum is Δp = pf - pi = m. (vf -vi). Units of impulse are N.s.
F.t = m. (vf -vi) or F = m. (vf -vi) / t
This can be derived from Newton’s second law that F = m.a = m. (vf -vi)/t.
Impact force : It is the impact force and not change in momentum that causes damage. If a car is traveling at 40 km/hr and strikes a brick wall and another similar car traveling at the same velocity collides with a haystack, it is obvious that the first car will have more damage. In both cases the car is brought to a halt, therefore change in momentum is the same. But collision with a brick wall brings the car to a halt in a fraction of a second, whereas the time of impact with a haystack is much longer. Therefore the force acting on the first car is larger. This is the same reason why an egg or a cup falling on a hard floor will break, but that falling on a cushion or carpet will not. The cushion increases the time of impact, reducing the force applied. Seat belts decrease the impact velocity and airbags increase time of impact and thus save the passenger from serious injury or fatality. A boxer rides with the punch and not into it, for the same reason, to increase the time of impact. If you swing and hit a wall, then it will hit you back with equal force and cause injury to your fist, but if you swing with the same velocity at a piece of paper, there is no injury; the reason is that the paper cannot hit you with a force greater than your force (Law of Action and Reaction). The force with which you hit is reduced due to increased time of impact. A bouncing collision causes more damage due to increased momentum change.
Collisions : are of three types. In all types, momentum and total energy are conserved.
1. Elastic collision : is one in which there is no loss of energy due to heat or sound or deformation. Kinetic energy is conserved in elastic collisions. Bodies separate after collision. Most atomic and molecular collisions, in which no chemical reaction takes place, are perfectly elastic. Perfectly elastic collisions do not occur in the large scale everyday life.
2. Inelastic collision : is one in which kinetic energy of the objects is NOT conserved. Bodies separate after collision. Most collisions we observe in everyday life are inelastic collisions, since some energy is always lost as heat and sound. For example, bouncing a tennis ball on the ground.
3. Completely Inelastic collision : is one in which the objects stick together after collision. Kinetic energy is NOT conserved. For example, railway carriages shunt together after colliding and moving as one unit, two skaters colliding and sticking to each other, football players tackling each other and moving together after colliding.
Strategy for solving problems of collisions in 1-D
1. Identify the objects in the system
2. If no net external force acts on the system, momentum is conserved.
3. Identify the defining event to determine momentum before and after the event. Example; collision or explosion
4. Calculate the momentum of each object in the system before and after the event. Take the sum of momentum of all objects before and after the event separately.
5. Equate the initial and final momentum and solve for the unknown variable.
pi = pf
Elastic & Inelastic Collisions
(u1 & u2 are initial velocities and v1 & v2 are final velocities)
m1u1 + m2 u2 = m1v1 + m2 v2
Completely Inelastic Collisions
m1u1 + m2 u2 = (m1 + m2) v
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