Relationship between masses based on Newton´s second law

Data

• Acceleration of the first body a₁ = 6 m/s²
• Acceleration of the second body a₂ = 9 m/s²
• Force acting on the first body F₁ 
• Force acting on the second body F₂

Resolution

The heading tells us the force acting on the first body and the force acting on the second one is the same, so:

$$F_1=F_2$$

The relation, m₁/m₂ we can find it applying Newton´s second law ( F=m·a ), and assuming the mass does not vary while the force is acting, we have:

$$m_1·a_1=m_2·a_2\Rightarrow \frac{m_1}{m_2}=\frac{a_2}{a_1}\Rightarrow \frac{m_1}{m_2}=\frac{9}{6}=\frac{3·\cancel{3}}{2·\cancel{3}}=\frac{3}{2}$$

In another way, we can say the mass of the first body is one time and a half the mass of the second body ( m₁=m₂·1.5 ).

Note: Observe that we worked throughout the whole problem by using the magnitudes of forces and acceleration, without taking into account its vector nature. Actually, because ${\overrightarrow{F}}_1={\overrightarrow{F}}_2$, the accelerations will have the same direction and we can use their magnitudes to find the relation requested.