Question

A certain force gives an object of mass ${m_1}$ an acceleration of $12.0m/{s^2}$ and an object of mass ${m_2}$ an acceleration of $3.30m/{s^2}$. What acceleration would the force give to an object of mass
(A) ${m_1} + {m_2}$
(B) ${m_2} - {m_1}$

Answer 1

There is a particular force that gives an object of mass ${m_1}$an acceleration of $12.0m/{s^2}$. The same force gives another object of mass ${m_2}$ an acceleration of $3.30m/{s^2}$. We have to find the acceleration produced by the same force on an object of mass ${m_1} + {m_2}$ and ${m_2} - {m_1}$.

Complete step by step solution:
Let us assume that the force is $F$.
The force can be written as,
$F = ma$
where $m$ stands for the mass of the object and $a$ stands for the acceleration.
From this, we can write the mass as,
$m = \dfrac{F}{a}$
Therefore, the mass ${m_1}$ can be written as,
${m_1} = \dfrac{F}{{{a_1}}}$
The mass ${m_2}$ can be written as,
${m_2} = \dfrac{F}{{{a_2}}}$
Now, the acceleration on the object of mass ${m_1} + {m_2}$ will be,
$a = \dfrac{F}{{{m_1} + {m_2}}}$
Substituting for ${m_1}$ and ${m_2}$
We get
$a = \dfrac{F}{{\left( {\dfrac{F}{{{a_1}}}} \right) + \left( {\dfrac{F}{{{a_2}}}} \right)}}$
This can be written as,
$a = \dfrac{1}{{\dfrac{1}{{{a_1}}} + \dfrac{1}{{{a_2}}}}} = \dfrac{{{a_1}{a_2}}}{{{a_1} + {a_2}}}$
Given that ${a_1} = 12.0m/{s^2}$ and ${m_2} = 3.30m/{s^2}$
Substituting, we get
$a = \dfrac{{12 \times 3.30}}{{12 + 3.30}} = 2.59m/{s^2}$
Therefore the acceleration produced by the force $F$ on an object of mass ${m_1} + {m_2} = 2.59m/{s^2}$
The acceleration produced by the force $F$ on an object of mass ${m_2} - {m_1}$ will be
$a = \dfrac{F}{{{m_2} - {m_1}}}$
Substituting for ${m_1}$ and ${m_2}$
We get
$a = \dfrac{F}{{\dfrac{F}{{{a_2}}} - \dfrac{F}{{{a_1}}}}}$
This can be written as,
$a = \dfrac{1}{{\dfrac{1}{{{a_2}}} - \dfrac{1}{{{a_1}}}}} = \dfrac{{{a_1}{a_2}}}{{{a_1} - {a_2}}}$
Substituting the values of ${a_1}$ and ${a_2}$ we get,
$a = \dfrac{{12 \times 3.30}}{{12 - 3.30}} = 4.55m/{s^2}$
The acceleration produced by a force $F$ on an object of mass ${m_2} - {m_1} = 4.55m/{s^2}$

Additional information:
Every moving body has a property due to its motion called momentum. It can be defined as the product of the mass of the object and its velocity.

Note: The relation between the force applied on an object and the acceleration produced by that object due to force is given by Newton’s second law of motion. The second law enables us to measure the force. According to the second law of motion, the net force acting on an object is directly proportional to the rate of change of momentum.