Learn first: Newton's laws
In some processes the forces are very complicated and act for a very short time: a collision, an explosion, a gunshot, a kick. It is almost impossible to know the force at every instant, so applying Newton’s laws directly is hard. But the idea of momentum and the law of conservation of momentum solve such problems in a line or two. Alongside energy, momentum is one of the main conserved quantities of mechanics; it is the key to rockets, jet engines, car safety and particle physics.
The momentum of a body is the product of its mass and velocity:
p⃗ = mv⃗, unit kg·m/s
In problems momentum is written as a projection on an axis: a body moving along the axis has positive momentum, one moving against it negative momentum.
Direction matters: a 0.2 kg ball that hits a wall at 5 m/s and bounces back at the same speed keeps the size of its momentum but reverses its direction: Δp = m(v − (−v)) = 2mv = 2 kg·m/s, not zero!
Newton’s second law F = ma = m·Δv/Δt can be rewritten as
F⃗·Δt = Δp⃗ = mv⃗₂ − mv⃗₁
F·Δt is the impulse of the force (N·s). The change in a body’s momentum equals the impulse of the force on it. This is a more general form of the second law (Newton himself stated the law in terms of the change of the quantity of motion). If the force varies, the impulse equals the area under the F(t) graph.
The same change of momentum can come from a large force over a short time or a small force over a long time. So:
A 0.5 kg ball hits a wall head-on at 10 m/s and rebounds at 8 m/s. The impact lasts 0.02 s. What average force does the wall exert on the ball?
When two bodies interact, Newton’s third law says they feel equal and opposite forces for the same time. So their momenta change by equal and opposite amounts, and the total does not change:
In a closed system the vector sum of momenta is constant: m₁v⃗₁ + m₂v⃗₂ = m₁v⃗₁′ + m₂v⃗₂′
A “closed” system is one with no external forces or with external forces that cancel. In practice the law is also used when:
Method: choose an axis → write the momentum of each body as a signed projection → write “before = after” → read the direction from the sign of the answer.
A 60 kg boy jumps horizontally from a 140 kg boat onto the shore at 2 m/s. Both were at rest. What speed does the boat get?
Real collisions lie between these two limits. Momentum is always conserved; kinetic energy only in elastic collisions.
A 2 kg trolley moving at 3 m/s hits a stationary 1 kg trolley and they move off together. Find the common speed and the heat produced.
If a body throws part of itself one way, it moves the other way: this is jet (reaction) propulsion. If the system starts at rest, 0 = m₁v₁ − m₂v₂. Rockets, jet aircraft, the recoil of a gun, a hose pushing back, squid and jellyfish all work this way. A rocket needs neither something to push against nor air, so it works in space.
A rocket engine ejects gas continuously; the faster the gas leaves relative to the rocket and the larger the share of fuel, the higher the rocket’s final speed. The theory of rocket motion was developed by K. E. Tsiolkovsky; the first artificial satellite (1957) and the first person in space (Yuri Gagarin, 1961) were launched by rockets.
A 10 g bullet leaves a 10 kg rifle at 600 m/s. What is the recoil speed?
A 3 kg shell flying horizontally at 20 m/s bursts into two pieces; a 1 kg piece flies backwards at 10 m/s. What is the velocity of the other piece?