Question

At, t = 0 force $\overrightarrow{\mathbf{F}}\mathbf{=}\left(-5.00\hat{i}+5.00\hat{j}+4.00\hat{k}\right)\mathbf{N}$begins to act on a 2.00 kg particle with an initial speed of 4.00 m / s . What is the particle’s speed when its displacement from the initial point is role="math" localid="1657949233150" $\overrightarrow{d}\mathbf{=}\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{00}\hat{\mathbf{i}}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{00}\hat{\mathbf{j}}\mathbf{+}\mathbf{7}\mathbf{.}\mathbf{00}\hat{\mathbf{k}}\mathbf{)}\mathbf{m}$?

Short answer

Speed of the particle is 6.63 m/s.

Step-by-step solution

Given information

It is given that,

1. Force is$\overrightarrow{F}=\left(-5.00\hat{\mathrm{i}}+5.00\hat{\mathrm{j}}+4.00\hat{\mathrm{k}}\right)\mathrm{N}$
2. Displacement is $\overrightarrow{d}=(2.00\hat{\mathrm{i}}+2.00\hat{\mathrm{j}}+7.00\hat{\mathrm{k}})\mathrm{m}$
3. Mass is given asm = 2.00 kg
4. Initial velocity is given as.$v_i=4.00\mathrm{m}/\mathrm{s}$

Determining the concept

The problem deals with the work done which is the fundamental concept of physics. Work is the displacement of an object when force is applied on it. First, findthework done bythedot product ofthegiven vectors, and then find the final velocity of the particle.

Formulae are as follow:

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

Where,

F istheforce, $v_f{v}_i$arethefinal and initial velocities, m isthemass, d isthedisplacement and W is the work done.

Determining the speed of the particle

Thework done can be found as,

$$\begin{gathered} W=\overrightarrow{F}.\overrightarrow{d} \\ W=\left(-5.00\hat{i}+5.00\hat{j}+4.00\hat{k}\right).\left(2.00\hat{i}+2.00\hat{j}+7.00\hat{k}\right) \\ W=28.0\mathrm{J} \end{gathered}$$

Using this value, find the velocity,

$$\begin{gathered} W=\frac{1}{2}m{v}_f^2-\frac{1}{2}m{v}_i^2 \\ 28.0=\frac{1}{2}(2.00)v_f^2-\frac{1}{2}(2.00){(4.00)}^{2}28.0=v_f^2-16 \\ {v}_f^2=28.0+16 \\ {v}_f=6.63\mathrm{m}/\mathrm{s} \end{gathered}$$

Hence, the speed of the particle is $v_f=6.63\mathrm{m}/\mathrm{s}$.