What is Work and Power - It's Introduction, Mathematical Expression, Examples, Conservation of Momentum

Introduction:

Imagine a tiny ball rolling toward you. You can easily stop it with your toe. It has very little momentum. A bowling ball rolling toward you at the same speed. You will not stop it. Why? Because it has more mass. A bicycle coming at you slowly. You could probably step aside and grab it to stop it. That same bicycle coming at you at top speed. You'd get out of it's way. Why? Because it has more speed. Momentum is measure of how difficult it is to stop a moving object. The more mass an object has and the faster it's moving, the more momentum it has.

The Law of Conservation of Momentum:
In a closed system with no external forces, the total momentum before an event like a collision or explosion is equal to the total momentum after the event. This law is incredibly powerful for predicting the outcomes of interactions.

e.g. Imagine two ice skaters pushing off each other. Before they push, they are stationary. Total momentum = 0. After they push, they fly apart in opposite directions. The larger skater moves slower. The smaller skater moves faster. Their momentum is equal in magnitude but opposite in direction. So when you add them together, the total momentum is still 0.

Momentum can't be created or destroyed; it can only be passed around. The total push given to one person is exactly equal to the push given to the other, just in the opposite direction. Momentum was conserved. This principle allows engineers to calculate the results of car crashes, astronomers to predict the motions of galaxies after they collide, and pool players to plan their shots.

In physics, momentum (p) is defined as the product of an object's mass and its velocity.

p = m * v

Where:
p = momentum (kg·m/s)
m = mass (kg)
v = velocity (m/s)

Because velocity is a vector, momentum is also a vector. An object with a momentum of 100 kg·m/s East is very different from one with 100 kg·m/s West.

People often confused between momentum and kinetic energy.

Momentum (p) is about how much push an object has and how hard it is to stop. While Kinetic Energy (KE) is about how much damage or destructive potential the motion can cause.

e.g. Imagine you are standing in a hallway, and bowling ball and basketball are rolled toward you. Your job is to stop them. A heavy bowling ball rolling slowly. While light basketball rolling very fast. Which one has more momentum. The bowling ball has more momentum. Even though it's slow, its huge mass gives it a lot of momentum. It would be very difficult to stop with your hands. You have to push back hard to slow over time. Momentum means how much force do I need to apply and for how long to stop this thing. Which one has more kinetic energy. The basketball has more kinetic energy. Its high speed is much more important than its light weight. If you tried to stop it, it would slam into your hands with a stinging force. It might even bounce off and cause something to fall over. Kinetic Energy means If this thing hits something, how much will it break, crush, or damage it.

e.g. Think of a bullet. Its momentum determines how far it will bury itself into a wall. how much it can push the wall. Its kinetic energy determines how much it will shatter, crack, or destroy the area around the hole.






Impulse:

If momentum is the amount of motion an object has, then impulse is the change in motion you give to it. The most important part of impulse is the time component. It’s not just how hard you push, but for how long you push.

Imagine you're pushing a child on a swing. There are two ways, how you want to push. First option is you give one giant, sudden push. This is a huge force for a very short time. The swing jerks forward violently. Second options is you push gently but keep your hands on their back, pushing for the entire way. This is a smaller force for a much longer time. The swing accelerates smoothly and goes just as high than the first option. In both cases, you delivered the same total impulse. You gave the swing the same total change in motion. You just did it in two different ways. One is big force for short time and second is small force for long time. Impulse is not just the force, it's how long you apply that force. To change that motion means to speed something up, slow it down, or change its direction, you need to apply an impulse.

Remember two things about impulse. Impulse is the average force applied to an object multiplied by the time period over which it is applied. Impulse is equal to the change in an object's momentum.

Impulse = Force × Time
= Change in Momentum
J = F × Δt = m × Δv

Where:
J = impulse (N·s)
F = average force (N)
Δt = time (s)
m = mass (kg)
Δv = change in velocity (m/s)

e.g.
1. Imagine about mechanism of car air bags during car crash. You are driving car at high speed. To stop your body's momentum i.e. change your motion from 100 mph to 0 mph. Imagine car without an airbag, in the absence of air bag, the steering wheel applies a massive force over a very short time to stop you. This impulse is deadly. While the airbag increases the time it takes to stop you. It cushions you and slows you down more gently. You are still going from 100 to 0 means same momentum. Same total impulse. if Time (t) goes up, the Force (F) must go down.

2. Imagine when you are going to catch a ball. When a ball hits your glove, it has momentum. If you keep your hand stiff short time, the ball stops suddenly and injures your hand with large force. If you pull your hand back as you catch which causes increasing the time, you slow the ball down more gently. The force is spread out, so it doesn't hurt.

3. Imagine kicking a soccer ball and kicking a wall. Your foot applies a force over a period of time (the time your foot is in contact with the ball). This impulse gives the ball momentum, and it flies away.

4. Imagine stopping a rolling bowling ball. Stop it instantly with a brick wall. The wall applies a massive force for a small amount of time. This is a sudden, impulse. There is another to stop it. Stop it slowly with your hands. You apply a gentle force for a longer amount of time. You slow it down gradually. In both cases, you delivered the same total impulse. You changed the ball's momentum from a certain value to zero. You just did it in two very different ways. Large force with short time and small force with long time.

People get confused about difference between force and impulse. Force (F) is just a push or a pull. While, Impulse (J = FΔt) is the effect of a force applied over time. It's what changes the momentum.