Newton's Third Law — Question 2

Question 2

Equal interaction forces can produce unequal accelerations

Two carts A and B of masses 2 kg and 5 kg interact on a frictionless line. At an instant A exerts +10 N on B. There are no other horizontal forces.

1. Find the force B exerts on A.

2. Find both accelerations.

3. Explain why a larger acceleration does not imply a larger interaction force.

Question 2 - Solution

Equal interaction forces can produce unequal accelerations

(a) Reverse the force vector.

$$\boxed{F_{B\to A}=-F_{A\to B}=-10\,\mathrm N}$$ B exerts -10 N on A. Equality and opposition hold for the mutual forces regardless of the masses.

(b) Divide by each body’s own mass.

$$a_A=-10/2=-5,\quad a_B=10/5=2\quad\mathrm{m/s^2}$$ $a_A=-10/2=-5$ m/s$^2$ and $a_B=10/5=2$ m/s$^2$.

(c) Distinguish force symmetry from response.

$$\frac{|a_A|}{|a_B|}=\frac{m_B}{m_A}=\frac52,\quad m_Aa_A+m_Ba_B=0$$ The force magnitudes are equal, while inertia differs. The lighter cart has a larger acceleration magnitude. Their acceleration ratio is inverse to their mass ratio, not evidence of an unequal action-reaction pair.