The acceleration of a particle starting from rest varies with time according to relation, . 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. $2\alpha {t}^{2}+\beta t$

Subtopic:  Integration |
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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

Subtopic:  Integration |
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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:

1. 30

2. 31

3. 32

4. 33

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

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

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

3. 6x+1

4. $3{x}^{2}+x$

Subtopic:  Integration |
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The current through a wire depends on time as i =(2+3t) A. The charge that crosses through the wire in 10 seconds is:
1. 150 C
2. 160 C
3. 170 C
4. None of there

Subtopic:  Integration |
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${\int }_{0}^{Q}\frac{q}{C}dq$, 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. $\frac{{Q}^{2}}{2C}$

Subtopic:  Integration |
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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 $\mathrm{F}=\left(3{\mathrm{t}}^{2}+2\mathrm{t}+5\right)\mathrm{N}$, 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

Subtopic:  Integration |
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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:

1. 250 J

2. 500 J

3. 100 J

4. 1000 J

Subtopic:  Integration |
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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 $F\left(x\right)=\left({x}^{2}-2x+1\right)N$, 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

Subtopic:  Integration |
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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. $-\mathrm{cos}\left(\mathrm{t}\right)+\frac{{\mathrm{t}}^{2}}{2}$

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

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

4. None of these

Subtopic:  Integration |
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