A particle starts from the origin at t=0 and moves in the x-y plane with constant acceleration 'a' in the y direction. Its equation of motion is $y=b{x}^2$. The x component of its velocity (at t=0) is:

1. variable

2. $\sqrt{\frac{2a}{b}}$

3. $\frac{a}{2b}$

4. $\sqrt{\frac{a}{2b}}$

The \(x\) and \(y\) coordinates of the particle at any time are \(x = 5t-2t^2\) and \(y=10t\) respectively, where \(x\) and \(y\) are in metres and \(t\) is in seconds. The acceleration of the particle at \(t=2\) s is:
1. \(0\) m/s²
2. \(5\) m/s²
3. \(-4\) m/s²
4. \(-8\) m/s²

A particle moves so that its position vector is given by \(r=\cos \omega t \hat{x}+\sin \omega t \hat{y}\) where \(\omega\) is a constant. Based on the information given, which of the following is true?

A body is moving with velocity 30 m/s towards east. After 10 s its velocity becomes 40 m/s towards north. The average acceleration of the body is

(1) $7<mspace\; width="1em"></mspace>\mathrm{m}/{\mathrm{s}}^2$                                             

(2) $\sqrt{7}<mspace\; width="1em"></mspace>\mathrm{m}/{\mathrm{s}}^2$

(3) $5<mspace\; width="1em"></mspace>\mathrm{m}/{\mathrm{s}}^2$                                             

(4) $1<mspace\; width="1em"></mspace>\mathrm{m}/{\mathrm{s}}^2$

A particle moves in space such that:
\(x=2t^3+3t+4;~y=t^2+4t-1;~z=2\sin\pi t\)
where \(x,~y,~z\)  are measured in meters and \(t\) in seconds. The acceleration of the particle at \(t=3\) seconds will be:

A particle moves along a parabolic path \(y =9x^2\) in such a way that the \(x\) component of the velocity remains constant and has a value of \(\frac{1}{3}~\text{m/s}\)${\mathrm{}}^{}$. It can be deduced that the acceleration of the particle will be:
1. \(\frac{1}{3}\hat j~\text{m/s}^2\)
2. \(3\hat j~\text{m/s}^2\)
3. \(\frac{2}{3}\hat j~\text{m/s}^2\)
4. \(2\hat j~\text{m/s}^2\)

If the position of a particle varies according to the equations \(x= 3t^2\)${\mathrm{}}^{}$, \(y =2t\), and \(z= 4t+4\), then which of the following is incorrect?

The position vector of a particle \(\overrightarrow r\) as a function of time \(t\) (in seconds) is \(\overrightarrow r=3 t \hat{i}+2t^2\hat j~\text{m}\). The initial acceleration of the particle is:
1. \(2~\text{m/s}^2\)
2. \(3~\text{m/s}^2\)
3. \(4~\text{m/s}^2\)
4. zero

At a certain instant, a particle moving in the \(xy\text-\)plane has a velocity of \(\vec v=(2\hat{i}+3\hat{j})~\text{m/s}\) and an acceleration of \(\vec a=(-3\hat{i}+2\hat{j})~\text{m/s}^2.\) What is the rate of change of the particle’s speed at that instant?
1. \(\sqrt{13}\) m/s²
2. \(-1\) m/s²
3. \(1\) m/s²
4. zero

A particle is moving along a curve. Select the correct statement.