Question

A constant force is applied to an object, causing the object to accelerate at 10 m/s². What will the acceleration be if

a. The force is halved?

b. The object’s mass is halved?

c. The force and the object’s mass are both halved?

d. The force is halved and the object’s mass is doubled?

Short answer

According to Newton's Second Law, the net force acting on an object is directly proportional to the acceleration of the object.

It is interpreted as the force acting on the object is equal to the product of mass and acceleration.

Step-by-step solution

Given solution:

The acceleration of the object at constant force F is given by a = 10 m/s²

Calculating the acceleration when the force is halved :

Since force is directly proportional to the acceleration of the object by Newton's Second law, it can be written as

$$\begin{gathered} F=ma \\ F\propto a \end{gathered}$$

Consider the initial force to be F and initial acceleration be a.

If the force F' is halved, then the acceleration a' is also halved.

Therefore, the new acceleration is calculated as :

$$\begin{gathered} \frac{F}{F^{\text{'}}}=\frac{a}{a^{\text{'}}} \\ \frac{F}{F/2}=\frac{10m/s^2}{a^{\text{'}}} \\ {a}^{\text{'}}=5m/s^2 \end{gathered}$$

Calculating the acceleration when the object’s mass is halved 

The formula to calculate the force acting on the object is given by F=ma

In terms of relation of acceleration and mass of the object, it can be written as

$$\begin{gathered} a=\frac{F}{m} \\ a\propto \frac{1}{m} \end{gathered}$$

Consider the original acceleration be a= 10 m/s² and mass be m.

When the mass is halved, let the final mass be m'=m/2 and the acceleration be a' respectively.

The new acceleration of the object when the mass is doubled is given by

$$\begin{gathered} \frac{a^{\text{'}}}{a}=\frac{m}{m^{\text{'}}} \\ {a}^{\text{'}}=\frac{m}{m/2}\times 10m/s^2 \\ =20m/s^2 \end{gathered}$$

Calculating the acceleration when the force and the object’s mass are both halved :

Consider the original force, F, mass be m and acceleration of the object be a=10 m/s².

The new force be F' =F/2 and mass be m'=m/2 respectively.

The new acceleration when the force and the object’s mass are both doubled can be calculated as

$$\begin{gathered} F=ma \\ a=\frac{F}{m} \\ \frac{a^{\text{'}}}{a}=\frac{{\displaystyle \raisebox{1ex}{$F^{\text{'}}$}\!\left/ \!\raisebox{-1ex}{$m\text{'}$}\right.}}{{\displaystyle \raisebox{1ex}{$F$}\!\left/ \!\raisebox{-1ex}{$m$}\right.}} \\ =\frac{{\displaystyle \raisebox{1ex}{$F/2$}\!\left/ \!\raisebox{-1ex}{$m/2$}\right.}}{{\displaystyle \raisebox{1ex}{$F$}\!\left/ \!\raisebox{-1ex}{$m$}\right.}} \\ {a}^{\text{'}}=a \\ =10m/s^2 \end{gathered}$$

Calculating the acceleration when the the force is halved and the object’s mass is doubled 

Consider the original force, F, mass be m and acceleration of the object be a=10 m/s².

The new force be F' =2F and mass be m'=m/2 respectively.

The new acceleration when the force is doubled and the object’s mass is halved can be calculated as:

$$\begin{gathered} F=ma \\ a=\frac{F}{m} \\ \frac{a^{\text{'}}}{a}=\frac{{\displaystyle \raisebox{1ex}{$F^{\text{'}}$}\!\left/ \!\raisebox{-1ex}{$m\text{'}$}\right.}}{{\displaystyle \raisebox{1ex}{$F$}\!\left/ \!\raisebox{-1ex}{$m$}\right.}} \\ =\frac{{\displaystyle \raisebox{1ex}{$(F/2)$}\!\left/ \!\raisebox{-1ex}{$\left({\displaystyle 2m}\right)$}\right.}}{{\displaystyle \raisebox{1ex}{$F$}\!\left/ \!\raisebox{-1ex}{$m$}\right.}} \\ {a}^{\text{'}}=a/4 \\ =10/4m/s^2 \\ =2.5m/s^2 \end{gathered}$$