Question

A $\mathbf{2}\mathbf{}\mathit{k}\mathit{g}$ball rolls off a $\mathbf{30}\mathbf{}\mathit{m}$-high cliff, and lands $\mathbf{25}\mathbf{}\mathit{m}$from the base of the cliff. Express the displacement and the gravitational force in terms of vectors and calculate the work done by the gravitational force. Note that the gravitational force is $\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{-}\mathit{m}\mathit{g}\mathbf{,}\mathbf{0}\mathbf{)}$, where g is a positive number$\mathbf{+}\mathbf{9}\mathbf{.}\mathbf{8}\mathbf{}\mathit{N}\mathbf{/}\mathit{k}\mathit{g}$.

Short answer

The displacement and the gravitational force in terms of vector are $25\hat{i}-30\hat{j}$ and $-(19.6\mathrm{N})\hat{j}$

respectively and the work done by the gravitational force is $588\mathrm{J}$

Step-by-step solution

Identification of the given data

The given data can be listed below as,

• The mass of the ball is, $2kg$.

• The height of the cliff is, $30m$.

• The ball lands from the base of the cliff is, $25m$.

• The gravitational force is, $(0,-mg,0)$.

Significance of law of conservation of energy for the ball

This law states that energy is neither created nor destroyed, it gets only converted to one form or another.

So, the work done will be equal to the product of the force applied and the displacement.

Determination of the gravitational force in terms of vector

Let $+y$be the upward direction of the ball and$+x$ direction is the landing area of the ball.

Height along the $+x$ axis and also the role="math" localid="1657726786348" $+y$ axis is $25\mathrm{m}$ and $30\mathrm{m}$ respectively.

So, the displacement of the ball in the vector form is expressed as,

$d-25\hat{i}-30\hat{j}$

Determination of the displacement in terms of vector

The gravitational force of the ball in terms of vector is expressed as,

$\stackrel{\bullet }{F}=-mg\hat{j}$

Here, $m$ is the mass of the ball and $g$ is the acceleration due to gravity.

Substitute all the values in the above expression.

role="math" localid="1657727095174" $\stackrel{\backslash \mathrm{mathscr}F}{F}=-(2\mathrm{kg})(9.8\mathrm{N}/\mathrm{kg})\hat{j}$

$=-(19.6\mathrm{N})\hat{j}$

Determination of the work done by the gravitational force

From the law of conservation of energy, the work done by the gravitational force is as follows,

$W=F^A·d$

Here,$\stackrel{n}{F}$is the gravitational force and $d$is the displacement of the ball.

Substituting all the values in the above equation.

$W=-19.6\mathrm{N}\times \hat{j}\times (-30\mathrm{m})\hat{j}$

$=588\mathrm{N}·\mathrm{m}\times \frac{1\mathrm{J}}{1\mathrm{N}·\mathrm{m}}$

$=588\mathrm{J}$

The displacement and the gravitational force in terms of vector are $25\hat{i}-30\hat{j}$ and $-(19.6\mathrm{N})\overline{j}$ respectively and the work done by the gravitational force is $588\mathrm{J}$.