The acceleration \(a\) (in ${\mathrm{ms}}^{-2}$) of a body, starting from rest varies with time \(t\) (in \(\mathrm{s}\)) as per the equation \(a=3t+4.\) The velocity of the body at time \(t=2\) \(\mathrm{s}\) will be:
1. \(10\) ${\mathrm{ms}}^{-1}$
2. \(18\) ${\mathrm{ms}}^{-1}$
3. \(14\) ${\mathrm{ms}}^{-1}$
4. \(26\) ${\mathrm{ms}}^{-1}$

A point moves in a straight line under the retardation a${\mathrm{v}}^2$. If the initial velocity is \(\mathrm{u},\) the distance covered in \(\mathrm{t}\) seconds is:
1. $\mathrm{aut}$
2. $\frac{1}{\mathrm{a}}\ln$ $\left(\mathrm{aut}\right)$
3. $\frac{1}{\mathrm{a}}\ln$ $\left(1+\mathrm{aut}\right)$
4. $\mathrm{a}$ $\ln$ $\left(\mathrm{aut}\right)$

A body starts from the origin and moves along the X-axis such that the velocity at any instant is given by $(4t^3-2t)$, where t is in sec and velocity in m/s. What is the acceleration of the particle, when it is 2 m from the origin ?

1. 28 m/s²

2. 22 m/s²

3. 12 m/s²

4. 10 m/s²

The initial velocity of a particle is u (at t = 0) and the acceleration f is given by at. Which of the following relation is valid 

1. $v=u+a{t}^2$

2. $v=u+a\frac{t^2}{2}$

3. $v=u+at$

4. v = u

The velocity of a body depends on time according to the equation $v=20+0.1t^2$. The body is undergoing 

1. Uniform acceleration

2. Uniform retardation

3. Non-uniform acceleration

4. Zero acceleration

The displacement of a particle is given by $\mathrm{y}=\mathrm{a}+\mathrm{bt}+{\mathrm{ct}}^2-{\mathrm{dt}}^4$. The initial velocity and acceleration are, respectively:

A body is thrown vertically upwards. If the air resistance is to be taken into account, then the time during which the body rises is: 

The acceleration of a particle is increasing linearly with time t as bt. The particle starts from the origin with an initial velocity of v₀. The distance travelled by the particle in time t will be:

(1) $v_{0}t+\frac{1}{3}b{t}^2$

(2) $v_{0}t+\frac{1}{3}b{t}^3$

(3) $v_{0}t+\frac{1}{6}b{t}^3$

(4) $v_{0}t+\frac{1}{2}b{t}^2$

The acceleration of a particle starting from rest varies with time according to the relation A = – aω²sinωt. The displacement of this particle at a time t will be:

1. $-\frac{1}{2}\left({\mathrm{a\omega }}^2\mathrm{sin\omega t}\right){\mathrm{t}}^2$

2. $\mathrm{a\omega sin\omega t}$

3. $\mathrm{a\omega cos\omega t}$

4. $\mathrm{asin\omega t}$

A particle is projected with velocity $v_0$ along x-axis. The deceleration of the particle is proportional to the square of the distance from the origin i.e., $a=\alpha x^2.$ The distance at which the particle stops is :

1.  $\sqrt{\frac{3v_0}{2\alpha }}$
2.  ${\left(\frac{{\displaystyle 3v_o}}{{\displaystyle 2\alpha }}\right)}^{\frac{1}{3}}$
3.  $\sqrt{\frac{3v{}_0{}^2}{2\alpha }}$
4.  ${\left(\frac{{\displaystyle 3v{}_0{}^2}}{{\displaystyle 2\alpha }}\right)}^{\frac{1}{3}}$