Question 1: Work, Energy and Power & Circular Motion

Question

A body of mass 1 kg moves along a straight line with a velocity $v=2 x^2$. The work done by the body during displacement from $x=0$ to 5 m is $\_\_\_\_$ J.

Accepted Answer

Work done (W) is defined as the integral of force (F) over a displacement (dx):
$$ \mathrm{W}=\int_{\mathrm{x}_{\mathrm{i}}}^{\mathrm{x}_{\mathrm{f}}} \mathrm{Fdx} $$
From Newton's Second Law, $\mathrm{F}=\mathrm{ma}$, and acceleration $\mathrm{a}=\frac{\mathrm{dv}}{\mathrm{dt}}$.
Using the chain rule, $\mathrm{a}=\frac{\mathrm{dv}}{\mathrm{dx}} \cdot \frac{\mathrm{dx}}{\mathrm{dt}}=\mathrm{v} \frac{\mathrm{dv}}{\mathrm{dx}}$.
$$ W=\int_{x_i}^{x_f}\left[m\left(v \frac{d v}{d x}ight)ight] d x $$
$$\Rightarrow $$ $W=\int_{v_i}^{v_f} m v d v=m \int_{v_i}^{v_f} v d v=m\left[\frac{v^2}{2}ight]_{v_i}^{v_f}$
$$\Rightarrow $$ $\mathrm{W}=\frac{1}{2} \mathrm{~m} \mathrm{v}_{\mathrm{f}}^2-\frac{1}{2} \mathrm{~m} \mathrm{v}_{\mathrm{i}}^2$
$$\Rightarrow $$ $$ \mathrm{W}=\Delta \mathrm{K} . \mathrm{E} . $$
The mass of the body is $\mathrm{m}=1 \mathrm{~kg}$
The velocity of body is given as function of position, $\mathrm{v}=2 \mathrm{x}^2$
Initial position is $\mathrm{x}_{\mathrm{i}}=0 \mathrm{~m}$
Final position is $\mathrm{x}_{\mathrm{f}}=5 \mathrm{~m}$
At $\mathrm{x}_{\mathrm{i}}=0$ :
$$ \mathrm{v}_{\mathrm{i}}=2(0)^2=0 \mathrm{~m} / \mathrm{s} $$
At $\mathrm{x}_{\mathrm{f}}=5$ :
$$ v_f=2(5)^2=2(25)=50 \mathrm{~m} / \mathrm{s} $$
On substituting these values into the Work-Energy equation :
$$ \mathrm{W}=\frac{1}{2} \mathrm{~m}\left(\mathrm{v}_{\mathrm{f}}^2-\mathrm{v}_{\mathrm{i}}^2ight) $$
$$\Rightarrow $$ $\mathrm{W}=\frac{1}{2} 	imes 1 	imes\left(50^2-0^2ight)$
$$\Rightarrow $$ $\mathrm{W}=\frac{1}{2} 	imes(2500)$
$$\Rightarrow $$ $$ \mathrm{W}=1250 \mathrm{~J} $$
Therefore, the work done by the body during displacement from $\mathrm{x}=0$ to $\mathrm{x}=5 \mathrm{~m}$ is 1250 J .
Hence, the correct option is (C).

Answer

0

Answer

250

Answer

1250

Answer

1000

Work, Energy and Power & Circular Motion

A body of mass 1 kg moves along a straight line with a velocity v=2 x^2. The work done by the body during displacement from x=0 to 5 m is \_\_\_\_ J.

Choose the correct answer: