What happens if two people push each other on skates?

In one line

The rule

Conservation of momentum: In a closed system, the total momentum before an event is equal to the total momentum after the event, if no external force acts on it.

Formula: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'

• m₁, m₂: masses of person 1 and 2 (in kg)
• v₁, v₂: their speeds before pushing (in m/s)
• v₁', v₂': their speeds after pushing (in m/s)

Let's understand it simply

Imagine you and your friend are standing on smooth skates, facing each other. The floor is very slippery, like ice, so there is almost no friction (nothing to slow you down). If you both push each other, you both move apart — you go one way, your friend goes the other. It’s like when you sit in a boat and push another boat away; both boats move apart. The heavier person moves slower, and the lighter person moves faster, but both move away from each other.

How it works

1. Both people start at rest (not moving), so total momentum is zero.
2. They push each other — this is a force that acts for a short time.
3. By Newton’s third law (every action has an equal and opposite reaction), the force you apply on your friend is equal and opposite to the force your friend applies on you.
4. Both start moving in opposite directions.
5. The total momentum (mass × speed) after pushing is still zero, because the pushes cancel each other out.
6. The lighter person moves away faster, the heavier person moves away slower, but in opposite directions.

The story behind it

The idea of momentum and its conservation was studied by scientists like Isaac Newton in the 17th century. Newton’s third law — “For every action, there is an equal and opposite reaction” — explains why both people move. The law of conservation of momentum was developed to explain how objects behave when they interact, like in collisions or pushes, especially when no outside force (like friction) is acting.

Where you see it in real life

• Two children on skateboards push each other: both roll away in opposite directions — their speeds depend on their weights.
• Rowing a boat: when you push water backwards with the oar, the boat moves forward — action and reaction.
• Playing carrom: when two coins hit, both move away — their speeds depend on their masses.
• A cricket bowler and batsman: when the bat hits the ball, both feel a force, but the ball moves much faster because it is lighter.
• Diwali rocket: the rocket pushes hot gas down, and the rocket goes up — both move in opposite directions.
• Two people pushing off each other in a swimming pool: both float away in opposite directions.

Why it matters

• This rule is the reason rockets work — they push gases out, and move up.
• It explains how vehicles move, how guns recoil, and how collisions work in sports and accidents.
• It is used in designing safety features in cars (airbags, crumple zones).
• Later chapters in physics, especially in class 11 and 12, build on momentum and its conservation for all types of collisions and explosions.

Worked example

Let’s say:

• Person A: 50 kg
• Person B: 70 kg
• Both start at rest (v₁ = v₂ = 0)
• After pushing, Person A moves at 2 m/s (v₁' = 2 m/s)

Find: Speed of Person B (v₂')

Given: m₁ = 50 kg, v₁ = 0, m₂ = 70 kg, v₂ = 0, v₁' = 2 m/s
Step 1: Write the rule: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'
Step 2: Substitute values: 0 + 0 = 50 × 2 + 70 × v₂'
Step 3: Simplify: 0 = 100 + 70 × v₂'
Step 4: Solve for v₂': 70 × v₂' = -100
Step 5: v₂' = -100 / 70 = -1.43 m/s
Step 6: Answer: Person B moves at 1.43 m/s in the opposite direction.

Common mistakes

• Thinking only the lighter person moves — both move, but the heavier one moves slower.
• Forgetting that the total momentum must be zero if both start at rest.
• Mixing up direction: the two people move in opposite directions, not the same.
• Ignoring friction: in real life, friction slows you down, but on skates or ice, it’s very small.

Remember it

When two people push each other on skates, both roll away — the lighter one moves faster, but both move in opposite directions to keep total momentum the same.