Imagine you're standing on perfectly smooth ice, wearing skates. You're completely still. Now, you push a heavy medicine ball away from you. What happens? You roll backward. The harder you push the ball, the faster you roll back.
That's the core intuition: you can't push something away without being pushed back yourself. The push you give the ball is matched by an equal push on you, in the opposite direction. This isn't a special property of ice or skates — it's a fundamental rule of how forces work in the universe.
The Hidden Quantity That Never Changes
Physicists call the "amount of motion" an object has its momentum. For everyday speeds, momentum is simple:
p=mv
Where m is mass (how much stuff) and v is velocity (speed with direction). Momentum is a vector — it cares about which way you're going.
A truck creeping forward has huge momentum (big mass, small speed). A bullet zipping through air has moderate momentum (tiny mass, huge speed). A parked car has zero momentum (speed is zero).
Now here's the key: in any isolated system (no outside forces), total momentum stays the same. Always. Before, during, and after any interaction.
The Precise Statement
Important
Law of Conservation of Momentum:
In a closed, isolated system (no external forces), the total vector momentum of the system remains constant over time.
Mathematically, for two objects that interact (collide, push apart, explode):
p1,initial+p2,initial=p1,final+p2,final
Or in terms of masses and velocities:
m1u1+m2u2=m1v1+m2v2
Where u means initial velocity and v means final velocity.
Why This Works: Newton's Third Law in Disguise
When you push the medicine ball, your hand exerts a force F on the ball. By Newton's Third Law, the ball exerts an equal and opposite force −F back on your hand. These forces act for the same time Δt.
Force times time equals impulse, which equals change in momentum:
FΔt=Δp
For you and the ball:
Ball's momentum change: +FΔt (ball goes forward)
Your momentum change: −FΔt (you go backward)
Add them: +FΔt+(−FΔt)=0
Total change is zero. Momentum is conserved because forces always come in equal-and-opposite pairs.
Note
This is why a rocket works in the vacuum of space. It throws exhaust backward (one momentum change), and the rocket itself moves forward (equal opposite momentum change). No air needed — just Newton's Third Law and conservation of momentum.
What This Law Does NOT Mean
It does NOT mean individual objects keep constant momentum. Only the total of all objects in the system stays constant. Individual momenta can change wildly.
It does NOT apply if external forces act. If friction, gravity from outside, or a wall stops something, momentum is not conserved for that system. (You can expand the system to include the Earth or the wall, and then momentum is conserved again.)
It does NOT require collisions to be elastic. Even in a messy, sticky, energy-losing collision, momentum is still perfectly conserved. Energy can be lost to heat or deformation, but momentum never disappears.
The gun and shell form an isolated system. Before firing, both are at rest, so the total momentum is zero. After firing, the forward momentum of the shell must equal the backward momentum of the gun.
Initial momentum:
pi=0
Final momentum (taking the shell's direction as positive):
When the gun fires, momentum is conserved: the forward momentum of the shell equals the backward momentum of the gun. The recoil speed works out to 0.016 m s−1.
Why conservation of momentum?
Before the trigger is pulled, the system (gun + shell) is at rest. The total momentum is zero. When the gun fires, internal forces between the gun and shell accelerate them in opposite directions, but no external horizontal force acts on the system. This means the total momentum must remain zero afterward—whatever momentum the shell gains forward, the gun must gain an equal amount backward.
This is the heart of recoil: momentum is conserved in isolated systems, and the explosion inside the gun is an internal force.
Step-by-step solution
Set up the initial state.
Both gun and shell are at rest, so the initial momentum is:
pinitial=0
Define the final momenta.
Let the shell move forward with velocity vs=+80 m s−1 (taking forward as positive). The gun recoils backward with velocity vg, which we expect to be negative.
The shell's momentum is:
ps=msvs=0.020×80=1.6 kg m s−1
The gun's momentum is:
pg=mgvg=100×vg
Apply conservation of momentum.
Total momentum after firing must equal total momentum before: