Q4DQ - University Physics with Modern Physics

Question

If it takes total work W to give an object a speed v and kinetic energy K, starting from rest, what will be the object’s speed (in terms of v) and kinetic energy (in terms of K) if we do twice as much work on it, again starting from rest?

Step-by-Step Solution

Answer

The speed of the object and kinetic energy are $\sqrt{2}v$ and 2K after twice the work.

Step 1: Describe the Work-Energy theorem

The work energy theorem tells that the work done on the body results in a change in its kinetic energy.

$$\begin{gathered} {\mathbf{W}}_{\mathbf{total}}\mathbf{=}{\mathbf{K}}_{\mathbf{2}}\mathbf{-}{\mathbf{K}}_{\mathbf{1}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{∆}\mathbf{K} \end{gathered}$$

Step 2: Determine the object’s speed and kinetic energy

From the Work-Energy theorem,

$$\begin{gathered} W=K{E}_f-K{E}_I \\ =\frac{1}{2}m{v}_f^2-\frac{1}{2}m{v}_i^2 \end{gathered}$$ 

 

The initial velocity of the object is zero. So, $v_i=0$.

 

Substitute 0 for $v_i$, and v for $v_f$ in the above equation.

$W=\frac{1}{2} m{v}^2\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}......\left(1\right)$ 

 

Substitute 0 for $v_i$, and  $v_1$ for $v_f$ in the above equation.

$W_1=\frac{1}{2}m{v}_1^2$

 

Here, the work done by the charged particle is doubled.

$$\begin{gathered} 2\mathrm{W}=\frac{1}{2}{\mathrm{mv}}_1^2 \\ 2\left(\frac{1}{2}{\mathrm{mv}}^2\right)=\frac{1}{2}{\mathrm{mv}}_1^2 \\ 2{\mathrm{v}}^2={\mathrm{v}}_1^2 \\ {\mathrm{v}}_1=\sqrt{2{\mathrm{v}}^2} \\ =\sqrt{2}\mathrm{v} \end{gathered}$$

 

Therefore, the speed of the object after twice the work is $\sqrt{2}v$.

 

The initial kinetic energy is given by,

$K=\frac{1}{2}m{v}^2$ 

 

The final kinetic energy is given by,

$$\begin{gathered} K\text{'}=\frac{1}{2}m{v}_1^2 \\ =\frac{1}{2}m{\left(\sqrt{2}v\right)}^2 \\ =m{v}^2 \\ =2K \end{gathered}$$ 

 

Therefore, the kinetic energy is doubled or 2K.