If ${\mathrm{a}}_{\mathrm{r}} \mathrm{and} {\mathrm{a}}_{\mathrm{t}}$ represent radial and tangential accelerations, then the motion of particle will be uniformly circular for :

(1) ${\mathrm{a}}_{\mathrm{r}}$ = 0, ${\mathrm{a}}_{\mathrm{t}}$ = 0

(2) ${\mathrm{a}}_{\mathrm{r}}$ = 0, ${\mathrm{a}}_{\mathrm{t}}$ $\ne$ 0

(3) ${\mathrm{a}}_{\mathrm{r}}$ $\ne$ 0, ${\mathrm{a}}_{\mathrm{t}}$ = 0

(4) ${\mathrm{a}}_{\mathrm{r}}$ $\ne$ 0, ${\mathrm{a}}_{\mathrm{t}}$ $\ne$ 0

If a particle is moving in a circular orbit with constant speed, then:

A body is projected horizontally with speed u from a tower of height h. The magnitude of the velocity with which it will collide with the ground will be: [g is the acceleration due to gravity]

1.  $\sqrt{2\mathrm{gh}}$

2.  $\mathrm{u} + \sqrt{2\mathrm{gh}}$

3.  ${\mathrm{u}}^2 + \sqrt{2\mathrm{gh}}$

4.  $\sqrt{{\mathrm{u}}^2 + 2\mathrm{gh}}$

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

Coordinates of a particle as a function of time \(t\) are \(x= 2t\),
\(y =4t\). It can be inferred that the path of the particle will be:

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 body started moving with an initial velocity of \(4\) m/s along the east and an acceleration \(1\) m/s² along the north. The velocity of the body just after \(4\) s will be?

A projectile thrown from the ground has horizontal range R. If velocity at the highest point is doubled somehow, the new range will be:

(1) 3 R

(2) 1.5 R

(3) $\sqrt{3}$ R

(4) 2 R

A particle is projected from the origin with velocity $\left(4\hat{\mathrm{i}} + 9\hat{\mathrm{j}}\right)$ m/s. The acceleration in the region is constant and -10$\hat{\mathrm{j}}$ $\mathrm{m}/{\mathrm{s}}^2$. The magnitude of velocity after one second is

(1) 8 m/s

(2) $\sqrt{15}$ m/s

(3) $\sqrt{17}$ m/s

(4) $\sqrt{27}$ m/s