Question

An object experiencing a constant force accelerates at $8\mathrm{m}/{\mathrm{s}}^2$. What will the acceleration of this object be if

a. The force is halved? Explain.

b. The mass is halved?

c. The force is halved and the mass is halved?

Short answer

(a) When the force is halved, acceleration will be $4m/s^2$.

(b) When the mass is halved, acceleration will be localid="1650728862326" $16m/s^2$.

(c) When both the force and mass is halved, acceleration will be $8m/s^2$.

Step-by-step solution

Part (a) Step-1 Given Information : 

Let us consider an object experiencing a constant force accelerates at 8m/s² .

Read the formula of Newton 's second law in terms of proportionality.

$$\begin{gathered} F=ma\Rightarrow F\alpha m,F\alpha a,\left(Propertional\right) \\ a=\frac{F}{m}\Rightarrow a\alpha \frac{1}{m},(InverslyPropertional) \end{gathered}$$

A force localid="1650730686345" $F$causes an object of mass m to accelerate at

$a_1=\frac{F}{m}=8m/s^2$..

Let $a_2$be the new acceleration.

Part (a) Step-2 Simplification : 

The acceleration is proportional to the force, so if the force is halved, assuming mass has not changed, the acceleration is also halved.

$\left(\frac{F}{2}\right)=m{a}_2\Rightarrow a_2=\left(\frac{F}{2m}\right)=\frac{a_1}{2}=4m/s^2\text{.}$

Part (b) Step-1 Given Information: 

Let us consider an object experiencing a constant force accelerates at 8m/s² .

Read the formula of Newton 's second law in terms of proportionality.

$$\begin{gathered} F=ma\Rightarrow F\alpha m,F\alpha a,\left(Propertional\right) \\ a=\frac{F}{m}\Rightarrow a\alpha \frac{1}{m},(InverslyPropertional) \end{gathered}$$

A force $F$causes an object of mass m to accelerate at

$a_1=\frac{F}{m}=8m/s^2$...

Let $a_2$be the new acceleration.

Part (b) Step-2 Simplification: 

Acceleration and mass are inversely proportional. Halving the mass (maintaining the same force) means that

$F=\left(\frac{m}{2}\right)a_2\Rightarrow a_2=2\left(\frac{F}{m}\right)=2a_1=16m/s^2$.

acceleration has doubled.

Part (c) Step-1 Given Information: 

Let us consider an object experiencing a constant force accelerates at 8m/s² .

Read the formula of Newton 's second law in terms of proportionality.

$$\begin{gathered} F=ma\Rightarrow F\alpha m,F\alpha a,\left(Propertional\right) \\ a=\frac{F}{m}\Rightarrow a\alpha \frac{1}{m},(InverslyPropertional) \end{gathered}$$

A force $F$causes an object of mass m to accelerate at

$a_1=\frac{F}{m}=8m/s^2$...

Let $a_2$be the new acceleration.

Part (c) Step-2 Simplification: 

If the force and mass are both halved, then the ratio $\frac{F}{m}=\frac{\frac{1}{2}F}{\frac{1}{2}m}$has not changed

role="math" localid="1650728485846" $\left(2F\right)=\left(2m\right)a_2$

$a_2=\frac{F}{m}=a=8m/s^2$.