Question

A body of mass m₁ exerts a force on another body of mass m₂. If the magnitude of the acceleration of m₂ is a₂, then the magnitude of the acceleration of m₁ is (considering only two bodies in space).

Answer 1

Step 1: Given data

1. Mass of the bodies are $M_1$ and $M_2$.
2. The acceleration of the body of mass $M_2$ is $a_2$.

Step 2: Acceleration and force

1. Acceleration is the time rate of change in velocity.
2. From Newton's second law, force is the product of mass and acceleration, i.e, $F=ma$.
3. From Newton's third law, for every action, there has an equal and opposite reaction.

Step 3: Finding the acceleration

Let $a_1$ be the acceleration of the body of mass $M_1$.

Now, the force exerted on $M_1$ is

$F_1=M_1{a}_1$

And the force exerted on $M_2$ is

$F_2=M_2{a}_2$

From Newton's third law of motion,

$\left|F_1\right|=\left|F_2\right|$

or $M_1{a}_1=$ $M_2{a}_2$

or $a_1=\frac{M_2{a}_2}{M_1}$

So, the magnitude of the acceleration of $M_1$ is ${\mathit{a}}_{\mathbf{1}}\mathbf{=}\frac{{\mathbf{M}}_{\mathbf{2}}{\mathbf{a}}_{\mathbf{2}}}{{\mathbf{M}}_{\mathbf{1}}}$