The position vector of a particle \(\vec{R }\) as a function of time \(t\) is given by:
\(\vec{{R}}=4 \sin (2 \pi {t}) \hat{i}+4 \cos (2 \pi {t}) \hat{j},\)
where \(R\) is in metres, \(t\) is in seconds and \({\hat{i},\hat{j}}\) denotes unit vectors along \({x}\) and \({y}\text-\)directions, respectively. Which one of the following statements is wrong for the motion of the particle?

1.	Acceleration is along \((\text{-}\vec R )\).
2.	Magnitude of the acceleration vector is \(\frac{v^2}{R}\), 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 \(4\) m.

A projectile is fired from the surface of the earth with a velocity of \(5\) ms^–1 and at an angle \(\theta\) with the horizontal. Another projectile fired from another planet with a velocity of \(3\) ms^–1 at the same angle follows a trajectory that is identical to 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 \(2\hat i+3\hat j~\)m/s. Its velocity (in m/s) at point \(B\) is:
              

A particle moves in a circle of radius \(5\) cm with constant speed and time period \(0.2\pi\) s. The acceleration of the particle is:

A body is moving with a velocity of \(30\) m/s towards the east. After \(10\) s, its velocity becomes \(40\) m/s towards the north. The average acceleration of the body is:
1. \( 7~\text{m/s}^2\)
2. \( \sqrt{7}~\text{m/s}^2\)
3. \(5~\text{m/s}^2\)
4. \(1~\text{m/s}^2\)

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:

A particle of mass m is projected with velocity v making an angle of 45° with the horizontal. When the particle lands on 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. \(30^\circ\)
2. \(45^\circ\)
3. \(60^\circ\)
4. \(0\)