A body starts moving from rest on a horizontal ground such that the position vector of the body with respect to its starting point is given by \(r= 2 t\hat{i}+3t^2\hat j\). The equation of the trajectory of the body is:
1. \(y =1.5x\)
2. \(y =0.75x^2\)
3. \(y =1.5x^2\)
4. \(y =0.45x^2\)

A particle starts moving with constant acceleration with initial velocity (\(\hat{\mathrm{i}}+5\hat{\mathrm{j}}\)$\stackrel{}{\mathrm{}}$) m/s. After \(4\) seconds, its velocity becomes (\(3\hat{\mathrm{i}}-2\hat{\mathrm{j}}\)) m/s. The magnitude of its displacement in 4 seconds is:
1. \(5\) m
2. \(10\) m
3. \(15\) m
4. \(20\) m

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

A projectile is projected with initial kinetic energy \(K\). If it has kinetic energy \(0.25K\) at its highest point, then the angle of projection is:
1. \(30^{\circ}\)
2. \(45^{\circ}\)
3. \(60^{\circ}\)
4. \(75^{\circ}\)

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

If three coordinates of a particle change according to the equations \(x = 3 t^{2}, y = 2 t , z= 4\), then the magnitude of the velocity of the particle at time \(t=1\) second will be:
1. \(2\sqrt{11}~\text{unit}\)
2. \(\sqrt{34}~\text{unit}\)
3. \(40~\text{unit}\)
4. \(2\sqrt{10}~\text{unit}\)

In a uniform circular motion, if the speed of the particle is \(2\) m/s and radius of the circle is \(2\) m, then the values of centripetal and tangential acceleration are, respectively:
1. \(2~\text{m/s}^2,~2~\text{m/s}^2\)
2. \(2~\text{m/s}^2,~1~\text{m/s}^2\)
3. \(0,~2~\text{m/s}^2\)
4. \(2~\text{m/s}^2,~0\)

A person, who can swim with speed \(u\) relative to water, wants to cross a river (of width \(d\) and water is flowing with speed \(v\)). The minimum time in which the person can do so is:
1. \(\frac{d}{v}\)
2. \(\frac{d}{u}\)
3. \(\frac{d}{\sqrt{v^{2}   +   u^{2}}}\)
4. \(\frac{d}{\sqrt{v^{2}   -   u^{2}}}\)

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: