The position vector of a particle R as a function of time t is given by 
$\overrightarrow{\mathrm{R}}=4\sin \left[2\mathrm{\pi t}\right]\hat{\mathrm{i}}+4\cos \left[2\mathrm{\pi t}\right]\hat{\mathrm{j}}$
Where R is in meters, t is in seconds and $\hat{\mathrm{i}}, \hat{\mathrm{j}}$ denote unit vectors along x and y-directions, respectively. Which one of the following statements is wrong for the motion of the particle?

1. Acceleration is along $-\overrightarrow{\mathrm{R}}$.

2. Magnitude of the acceleration vector is $\frac{{\mathrm{v}}^2}{\mathrm{R}}$, where v is the velocity of the particle.

3. Magnitude of the velocity of the particle is 8 m/s.

4. Path of the particle is a circle of radius 4m.

A projectile is fired from the surface of the earth with a velocity of 5ms^–1 and at an angle θ with the horizontal. Another projectile fired from another planet with a velocity of 3ms^–1 at the same angle follows a trajectory which is identical with the trajectory of the projectile fired from the earth. The value of the acceleration due to gravity on the other planet is: (given g = 9.8 ms^–2)

1. 3.5 m/s²

2. 5.9 m/s²

3. 16.3 m/s²

4. 110.8 m/s²

The velocity of a projectile at the initial point A is $\left(2\hat{\mathrm{i}}+3\hat{\mathrm{j}}\right)$m/s. Its velocity (in m/s) at point B is:
            

1. $-2\hat{\mathrm{i}}+3\hat{\mathrm{j}}$

2. $2\hat{\mathrm{i}}-3\hat{\mathrm{j}}$

3. $2\hat{\mathrm{i}}+3\hat{\mathrm{j}}$

4. $-2\hat{\mathrm{i}}-3\hat{\mathrm{j}}$

The horizontal range and the maximum height of a projectile are equal. The angle of projection of the projectile is:

1. $\theta ={\tan }^{-1}\left(\frac{1}{4}\right)$

2. $\theta ={\tan }^{-1}\left(4\right)$

3. $\theta ={\tan }^{-1}\left(2\right)$

4. $\theta ={45}^0$

A particle moves in a circle of radius 5 cm with constant speed and time period $0.2\mathrm{\pi }$ sec. The acceleration of the particle is:

1.  25 m/s²

2.  36 m/s²

3.  5 m/s²

4.  15 m/s²

A body is moving with a velocity of 30 m/s towards the east. After 10 sec, its velocity becomes 40 m/s towards the 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 missile is fired for a maximum range with an initial velocity of 20 m/s. If g= 10 m/s², then the range of the missile will be

1.  50 m

2.  60 m

3.  20 m

4.  40 m

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

1. 7 units

2. 7$\sqrt{2}$units

3. 8.5 units

4. 10 units

A particle of mass m is projected with velocity v making an angle of 45° with the horizontal. When the particle lands on the level ground, the magnitude of change in its momentum will be:

1.  $2\mathrm{mv}$

2.  $\mathrm{mv}/\sqrt{2}$

3.  $\mathrm{mv}\sqrt{2}$

4.  zero

A particle starting from the origin (0, 0) moves in a straight line in the (x, y) plane. Its coordinates at a later time are ($\sqrt{3}$, 3). The path of the particle makes an angle of .......................with the x-axis :

1. 30^o

2. 45^o

3. 60^o

4. 0^o