A projectile is given an initial velocity of \(\hat i+2\hat j\). The cartesian equation of its path is: (Take \(g = 10 \ \text{ms}^{ - 2}\)) 

1. \(y=2x-5x^2\)

2. \(y=x-5x^2\) $\mathrm{}$

3. \(4y=2x-5x^2\)$\mathrm{}$

4. \(y=2x-25x^2\)

Ship A is moving westward with a speed of \(10\ \text{km h}^{- 1}\) and Ship B, south of A, is moving northwards with a speed of \(10\ \text{km h}^{- 1}\)${\mathrm{}}^{}$. The time after which the distance between them becomes the shortest is:

1. \(5\) hr

2. \(5 \sqrt2\) hr

3. \(10 \sqrt2\)$\mathrm{}$ hr

4. \(0\) hr

A particle projected with kinetic energy ${\mathrm{k}}_0$ with an angle of projection $\mathrm{\theta }$. Then the variation of kinetic K with vertical displacement y is

1.  linear

2.  parabolic

3.  hyperbolic

4.  periodic

A particle starts from the origin at t=0 and moves in the x-y plane with constant acceleration 'a' in the y direction. Its equation of motion is $y=b{x}^2$. The x component of its velocity (at t=0) is:

1. variable

2. $\sqrt{\frac{2a}{b}}$

3. $\frac{a}{2b}$

4. $\sqrt{\frac{a}{2b}}$

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

1. 7 units

2. $7\sqrt{2}$ units

3. 8.5 units

4. 10 units

The gravity in space is given by $\overrightarrow{\mathrm{g}}=-10<mspace\; width="1em"></mspace>\hat{j}<mspace\; width="1em"></mspace>m{s}^{-2}$. Two particles are simultaneously projected with velocity $\overrightarrow{{\mathrm{u}}_1}=\left(10<mspace\; width="1em"></mspace>m{s}^{-1}\right)\hat{i}+\left(10<mspace\; width="1em"></mspace>m{s}^{-1}\right)\hat{j}$ and $\overrightarrow{{\mathrm{u}}_2}=\left(20<mspace\; width="1em"></mspace>m{s}^{-1}\right)\hat{i}+\left(10<mspace\; width="1em"></mspace>m{s}^{-1}\right)\hat{j}$. Then, the ratio of their times of flight

1. 1:1

2. 1:2

3. 2:1

4. none

A train is moving towards the east and a car is along the north at the same speed. The observed direction of the car to the passenger on the train is:

Two racing cars of masses \(m_1\) and \(m_2\) are moving in circles of radii \(r_1\) and \(r_2\) respectively. Their speeds are such that each makes a complete circle in the same duration of time \(t\). The ratio of the angular speed of the first to the second car is:

A particle P is moving in a circle of radius ‘a’ with a uniform speed v. C is the centre of the circle and AB is a diameter. When passing through B the angular velocity of P about A and C are in the ratio 

1. 1 : 1

2. 1 : 2

3. 2 : 1

4. 4 : 1

Two bodies of mass 10 kg and 5 kg moving in concentric orbits of radii R and r such that their periods are the same. Then the ratio between their centripetal acceleration is 

1. R/r

2. r/R

3. R²/r²

4. r²/R²