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, \(v=\frac{dx}{dt}\))

1. vary with time.

2. be independent of time.

3. be inversely proportional to time.

4. be inversely proportional to acceleration.

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 sec will be: (Here, \(a=v\frac{dv}{dx}\))

1.  0.05 m/s

2. 0.5 m/s

3. 5 m/s

4. 50 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 sec will be: (\(Here, a=\frac{dv}{dt}~ and~v=\frac{dx}{dt}\))

1. 6 m/s, 4 m

2. 4 m/s, 6 m

3. 3 m/s, 2 m

4. 2 m/s, 3 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. $\frac{-k}{a}$

Consider a liquid of density $\rho$ in a container that spins with angular velocity $\omega$ as shown in figure. Find relation between y and x for any point P, if liquid rises due to rotation. [This question is only for Dropper and XII batch]

1. $y=\frac{\omega x}{2g}$

2. $y=\frac{{\omega }^2{x}^2}{2g}$

3. $y=\frac{{\omega }^{2}x}{2g}$

4. $y=\frac{\omega x^2}{2g}$