The displacement of the particle is zero at $t=0$ and at $t=t$ it is $x$. It starts moving in the $x$-direction with a velocity that varies as $v = k \sqrt{x}$, where $k$ is constant. The velocity will: (Here, $v=\frac{dx}{dt}$)

The acceleration of a particle is given as $a= 3x^2$.  At $t=0,v=0$ and $x=0$. It can then be concluded that the velocity at $t=2~\text{s}$ will be: (Here, $a=v\frac{dv}{dx}$)
1. $0.05~\text{m/s}$
2. $0.5~\text{m/s}$
3. $5~\text{m/s}$
4. $50~\text{m/s}$

The acceleration of a particle is given by $a=3t$ at $t=0$, $v=0$, $x=0$. The velocity and displacement at $t = 2~\text{sec}$ will be:
$\left(\text{Here,} ~a=\frac{dv}{dt}~ \text{and}~v=\frac{dx}{dt}\right)$
1. $6~\text{m/s}, 4~\text{m}$
2. $4~\text{m/s}, 6~\text{m}$
3. $3~\text{m/s}, 2~\text{m}$
4. $2~\text{m/s}, 3~\text{m}$

The 9 kg block is moving to the right with a velocity of 0.6 m/s on a horizontal surface when a force F, whose time variation is shown in the graph, is applied to it at time t = 0. Calculate the velocity v of the block when t= 0.4s. The coefficient of kinetic fricton is ${\mu }_k=0.3$. [This question includes concepts from Work, Energy & Power chapter]

1. 0.6 m/s

2. 1.2 m/s

3. 1.8 m/s

4. 2.4 m/s

The relationship between force and position is shown in the figure given (in one dimensional case). Find the work done by the force in displaying a body from x= 1 cm to x= 5cm is [This question includes concepts from Work, Energy and Power chapter]

1. 10 erg

2. 20 erg

3. 30 erg

4. 40 erg

A long spring is stretched by 2 cm, its potential energy is U. If the spring is streched by 10 cm, find the potential energy stored in it.  

1. 10 U

2. 15 U

3. 20 U

4. 25 U

A spring of spring constant $5\times {10}^3 N/m$ is stretched initially by 5 cm from the unstretched position. Find the work required to stretch it further by another 5 cm is   -

1. 15 J

2. 18.75 N .m

3. 20 J

4. 22.75 N .m

A constant force F is applied on a body. The power (P) generated is related to the time elapsed (t) as [This question includes concepts from Work, Energy and Power chapter]

1. $P \propto t^2$

2. $P \propto t$

3. $P \propto \sqrt{ t}$

4. $P \propto t^{3/2}$

The gravitational field due to a mass distribution is given by $\overrightarrow{I}= \frac{k}{x^2}\hat{i}$, where k is a constant. Assuming the potential to be zero at infinity, find the potential at a point x = a.[This question includes concepts from Gravitation chapter]

1. $\frac{k}{a^2}$

2. $\frac{-k}{a^2}$

3. $\frac{k}{a}$

4. $\frac{-k}{a}$