For any arbitrary motion in space, which of the following relations is true?

1.	\(\overrightarrow{v}_{\text {avg }}=\left(\frac{1}{2}\right)\left[\overrightarrow{v}\left(t_1\right)+\overrightarrow{v}\left(t_2\right)\right]\)
2.	\(\overrightarrow{v}(t)=\overrightarrow{v}(0)+\overrightarrow{a} t\)
3.	\(\overrightarrow{r}({t})=\overrightarrow{r}(0)+\overrightarrow{v}(0){t}+\frac{1}{2} \overrightarrow{a}{t}^2\)
4.	\(\overrightarrow{v}_{\text {avg }}=\frac{\left[\overrightarrow{r}\left(t_2\right)-\overrightarrow{r}\left(t_1\right)\right]}{\left(t_2-t_1\right)}\)

A particle is moving along a circle such that it completes one revolution in \(40\) seconds. In \(2\) minutes \(20\) seconds, the ratio of \(|displacement| \over distance\) will be:
1. \(0\)
2. \(\frac{1}{7}\)
3. \(\frac{2}{7}\)
4. \(\frac{1}{11}\)

Consider the motion of the tip of the second hand of a clock. In one minute (assuming \(R\) to be the length of the second hand), its:

A particle projected from origin moves in the \(x\text-y\) plane with a velocity \(\overrightarrow{v} = 3 \hat{i} + 6 x \hat{j}\)$\stackrel{}{\mathrm{}}$, where \(\hat i\) and \(\hat j\) are the unit vectors along the \(x\) and \(y\text-\)axis. The equation of path followed by the particle is:
1. \(y=x^2\)
2. \(y=\frac{1}{x^2}\)
3. \(y=2x^2\)
4. \(y=\frac{1}{x}\)

The position coordinates of a projectile projected from ground on a certain planet (with no atmosphere) are given by
\(y =4 t - 2 t^{2}~ \text{m}\)  and \(x =3t\) metre, where \(t\) is in seconds and point of projection is taken as the origin. The angle of projection of projectile with vertical is:
1. \(30^{\circ}\)
2. \(37^{\circ}\)
3. \(45^{\circ}\)
4. \(60^{\circ}\)

The velocity at the maximum height of a projectile is \(\frac{\sqrt{3}}{2}\) times its initial velocity of projection \((u)\). Its range on the horizontal plane is:
1. \(\frac{\sqrt{3} u^{2}}{2 g}\)
2. \(\frac{3 u^{2}}{2 g}\)
3. \(\frac{3 u^{2}}{ g}\)
4. \(\frac{u^{2}}{2 g}\)

The equation of a projectile is \(y = ax -bx^{2}\). Its horizontal range is?
1. \(\frac{a}{b}\)
2. \(\frac{b}{a}\)
3. \(a+b\)
4. \(b-a\)

When a particle is projected at some angle to the horizontal, it has a range \(R\) and time of flight \(t_1\). If the same particle is projected with the same speed at some other angle to have the same range, its time of flight is \(t_2\), then:
1. \(t_{1} + t_{2} = \frac{2 R}{g}\)
2. \(t_{1} - t_{2} = \frac{R}{g}\)
3. \(t_{1} t_{2} = \frac{2 R}{g}\)
4. \(t_{1} t_{2} = \frac{R}{g}\)

A car is moving at a speed of \(40\) m/s on a circular track of radius \(400\) m. This speed is increasing at the rate of \(3\) m/s². The acceleration of the car is:
1. \(4\) m/s²
2. \(7\) m/s²
3. \(5\) m/s²
4. \(3\) m/s²

A particle starts from the origin at \(t=0\) sec with a velocity of \(10\hat j~\text{m/s}\) and moves in the \(x\text-y\) plane with a constant acceleration of \((8.0\hat i +2.0 \hat j)~\text{m/s}^2\). At what time is the \(x\text-\)coordinate of the particle \(16\) m?