A particle moves in space such that

$\mathrm{x}=2{\mathrm{t}}^3+3\mathrm{t}+4 ; \mathrm{y}={\mathrm{t}}^2+4\mathrm{t}-1 ; \mathrm{z}=2 \sin \mathrm{\pi t}$

Where x, y, z are measured in metre and t in second. The acceleration of the particle at t=3s is

(A)  $36\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}} {\mathrm{ms}}^{-2}$

(B)  $36\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\mathrm{\pi }\hat{\mathrm{k}} {\mathrm{ms}}^{-2}$

(C)  $36\hat{\mathrm{i}}+2\hat{\mathrm{j}} {\mathrm{ms}}^{-2}$

(D)  $12\hat{\mathrm{i}}+2\hat{\mathrm{j}} {\mathrm{ms}}^{-2}$

The coordinates of a moving particle at a time \(t\), are given by, \(x= 5\sin 10t, y = 5\cos 10t\). It can be deduced that the speed of the particle will be:

If \(\overrightarrow {A}= \overrightarrow{B}+ \overrightarrow{C}\) and the magnitudes of \(\overrightarrow {A}, \overrightarrow {B},\overrightarrow {C}\) are \(5,4\) and \(3\) units, respectively. Then the angle between \(\overrightarrow {A}~\text{and}~\overrightarrow{C}\)$\stackrel{}{}$is:
1. \(\cos^{-1}\left(\frac{3}{5}\right)\)
2. \(\cos^{-1}\left(\frac{4}{5}\right)\)
3. \(\sin^{-1}\left(\frac{3}{4}\right)\)
4. \(\frac{\pi}{2}\)

Two particles are projected simultaneously from the level ground as shown in figure. They may collide after a time :

1. $\frac{{\mathrm{xsin\theta }}_2}{{\mathrm{u}}_1}$

2. $\frac{{\mathrm{xcos\theta }}_2}{{\mathrm{u}}_1}$

3. $\frac{{\mathrm{xsin\theta }}_2}{{\mathrm{u}}_1\sin \left({\mathrm{\theta }}_2-{\mathrm{\theta }}_1\right)}$

4. $\frac{{\mathrm{xsin\theta }}_1}{{\mathrm{u}}_2\sin \left({\mathrm{\theta }}_2-{\mathrm{\theta }}_1\right)}$

A particle is moving along the path y = ${\mathrm{x}}^2$, from x = 0m to x = 2m. Then the distance traveled by the particle is:

1.  4m

2.  $\sqrt{20}\mathrm{m}$

3.  $>\sqrt{20}\mathrm{m}$

4.  $<\sqrt{20}\mathrm{m}$

What determines the nature of the path followed by a particle:
1. Speed
2. Velocity
3. Acceleration
4. Both (2) and (3)

A stone is just released from the window of a train moving along a horizontal straight track. The stone will hit the ground following

(1) Straight path

(2) Circular path

(3) Parabolic path

(4) Hyperbolic path

The speed of a projectile at its maximum height is half of its initial speed. The angle of projection is:

1. $60°$                                         2. $15°$

3. $30°$                                          4. $45°$

A particle has initial velocity $\left(3\hat{i}+4\hat{j}\right)$ and has acceleration $\left(0.4\hat{i}+0.3\hat{j}\right)$. Its speed after 10s is 

(1) 7unit                                                         

(2) 7$\sqrt{2}$unit

(3) 8.5 unit                                                     

(4) 10 unit

A projectile is fired at an angle of 45$°$ with the horizontal. The elevation angle of the projectile at its highest point as seen from the point of projection is:

1. 60$°$                                           2. ${\tan }^{-1}\left(\frac{1}{2}\right)$

3. ${\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right)$                               4. $45°$