Equal forces, unequal accelerations

The idea

The forces two interacting objects exert on each other are equal in magnitude but generally produce unequal accelerations, because acceleration = force ÷ mass and their masses may differ.

Gun and bullet

When a gun fires a bullet, the bullet shoots forward and the gun is pushed back. The two forces are equal.

Do you expect the gun and the bullet to speed up equally?

• No, the lighter one speeds up more
• Yes, exactly equally
• The gun speeds up more

Two carts

Change the masses and compare the accelerations.

Same force, different mass

Both carts felt the same force, but acceleration is force divided by mass. The lighter cart had the larger acceleration. Equal masses gave equal accelerations.

The forces two interacting objects exert on each other are equal in magnitude but generally produce unequal accelerations, because acceleration = force ÷ mass and their masses may differ.

Notes

The two forces are equal, but the accelerations are generally unequal, since a = F/m and the masses may differ.

Check yourself

A gun and a bullet push each other with equal forces. Which has the larger acceleration?

A 5 kg gun is pushed back by a force of 2 N. What is its acceleration?

Answer: 0.4 m s⁻²

a = F/m = 2 N ÷ 5 kg = 0.4 m s⁻².

The 0.1 kg bullet is pushed by a force of 2 N. What is its acceleration?

Answer: 20 m s⁻²

a = F/m = 2 N ÷ 0.1 kg = 20 m s⁻².

Why do two interacting objects usually have unequal accelerations although the forces on them are equal?

• The gun, because its mass is larger. A larger mass gives a smaller acceleration for the same force, a = F/m.
• The bullet, because its mass is smaller — correct. Yes!
• Both have the same acceleration. The forces are equal, but the masses differ, so the accelerations differ.
• The forces are not really equal. The forces are equal in magnitude, as the third law says.
• The forces act in the same direction. The forces are opposite in direction.
• Their masses may be different — correct. Yes!