The acceleration of a particle starting from rest varies with time according to relation, $a=\alpha t+\beta$. The velocity of the particle at time instant t is: (\(Here, a=\frac{dv}{dt}\))

1. $\alpha t^2+\beta t$

2. $\alpha t^2+\frac{\beta t}{2}$

3. $\frac{\alpha t^2}{2}+\beta t$

4. $2\alpha t^2+\beta t$

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

Given velocity v(t)=$\frac{5}{2}\mathrm{t}+3$. Assume s(t) is measured in meters and t is measured in seconds. If s(0)=0, the position s(4) at t=4s is:  $\left(\mathrm{Given}, \mathrm{v}=\frac{\mathrm{ds}}{\mathrm{dt}}\right)$

1. 30

2. 31

3. 32

4. 33

If the force on an object as a function of displacement is $F\left(x\right)=3x^2+x$, what is work as a function of displacement $w\left(x\right) \left(W=\int \left(fdx\right)\right)$. Assume W(0)=0 and force is in the direction of the object's motion.

1. $\frac{3x^3}{2}+x^2$

2. $x^3+\frac{x^2}{2}$

3. 6x+1

4. $3x^2+x$

The current through a wire depends on time as i =(2+3t) A. The charge that crosses through the wire in 10 seconds is: $\left(\mathrm{Instantaneous} \mathrm{current}, \mathrm{i}=\frac{\mathrm{dq}}{\mathrm{dt}}\right)$
1. 150 C
2. 160 C
3. 170 C
4. None of there

${\int }_0^Q\frac{q}{C}dq$, where C is a constant, can be expressed as:

1. $\frac{Q^2}{C}$

2. $\frac{-Q^2}{2C}$

3. $\frac{-Q^2}{C}$

4. $\frac{Q^2}{2C}$

The impulse due to a force on a body is given by I=$\int \mathrm{Fdt}$. If the force applied on a body is given as a function of time (t) as $\mathrm{F}=(3{\mathrm{t}}^2+2\mathrm{t}+5)\mathrm{N}$, then impulse on the body between t=3 sec to t = 5 sec is:

1. 175 kg-m/sec

2. 41 kg-m/sec

3. 216 kg-m/sec

4. 124 kg-m/sec

The work done by gravity exerting an acceleration of -10 m/${\mathrm{s}}^2$ for a 10 kg block down 5 m from its original position with no initial velocity is:

$({\mathrm{F}}_{\mathrm{grav}}=\mathrm{mass} \times \mathrm{acceleration} \mathrm{and} \mathrm{W}={\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{F}(\mathrm{x}\left)\mathrm{dx}\right)$

1. 250 J

2. 500 J

3. 100 J

4. 1000 J

Work done by a force (F) in displacing a body by dx is given by W=$\int F\left(x\right).dx$. If the force is given as a function of displacement (x) by $F\left(x\right)=(x^2-2x+1)N$, then work done by the force from x=0 to x=3 m is:

1. 3 J

2. 6 J

3. 9 J

4. 21 J

If acceleration of a particle is given as a(t)=sin(t)+2t. Then the velocity of the particle will be:

(acceleration $\mathrm{a}=\frac{\mathrm{dv}}{\mathrm{dt}}$)

1. $-\cos \left(\mathrm{t}\right)+\frac{{\mathrm{t}}^2}{2}$

2. $-\sin \left(\mathrm{t}\right)+{\mathrm{t}}^2$

3. $-\cos \left(\mathrm{t}\right)+{\mathrm{t}}^2$

4. None of these