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. \(0\)
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) $7 \mathrm{m}/{\mathrm{s}}^2$                                             

(2) $\sqrt{7} \mathrm{m}/{\mathrm{s}}^2$

(3) $5 \mathrm{m}/{\mathrm{s}}^2$                                             

(4) $1 \mathrm{m}/{\mathrm{s}}^2$

A particle moves in space such that:
\(x=2t^3+3t+4;~y=t^2+4t-1;~z=2sin\pi t\)
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 = 9x² in such a way that the x component of the velocity remains constant and has a value of $\frac{1}{3}$ ${\mathrm{ms}}^{-1}$. It can be deduced that the acceleration of the particle will be:

1. $\frac{1}{3}$ $\hat{\mathrm{j}}$ ${\mathrm{ms}}^{-2}$

2. $3$ $\hat{\mathrm{j}}$ ${\mathrm{ms}}^{-2}$

3. $\frac{2}{3}$ $\hat{\mathrm{j}}$ ${\mathrm{ms}}^{-2}$

4. $2$ $\hat{\mathrm{j}}$ ${\mathrm{ms}}^{-2}$

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

The position vector of a particle $\overrightarrow{\mathrm{r}}$ as a function of time t (in seconds) is $\overrightarrow{\mathrm{r}}$ $=$ $\left(3\mathrm{t}\right)\hat{\mathrm{i}}$ $+$ $\left(2{\mathrm{t}}^2\right)\hat{\mathrm{j}}$ $\mathrm{m}$. The initial acceleration of the particle is:

1. 2 $\mathrm{m}/{\mathrm{s}}^2$

2. 3 $\mathrm{m}/{\mathrm{s}}^2$

3. 4 $\mathrm{m}/{\mathrm{s}}^2$

4. Zero

At an instant, the velocity and acceleration of a particle moving in the XY plane are $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}}$ m/s and $-3\hat{\mathrm{i}}+2\hat{\mathrm{j}}$ $\mathrm{m}/{\mathrm{s}}^2$. Rate of change of speed of the particle at this instant is:

(1) $\sqrt{13}$ $\mathrm{m}/{\mathrm{s}}^2$

(2) -1 $\mathrm{m}/{\mathrm{s}}^2$

(3) 1 $\mathrm{m}/{\mathrm{s}}^2$

(4) Zero

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