A particle is projected with a velocity u making an angle $\mathrm{\theta }$ with the horizontal. At any instant, its velocity v is at right angles to its initial velocity u; then v is:

1.  ucos$\mathrm{\theta }$

2.  utan$\mathrm{\theta }$

3.  ucot$\mathrm{\theta }$

4.  usec$\mathrm{\theta }$

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

1. $\mathrm{y}=2\mathrm{x}-5{\mathrm{x}}^2$

2. $\mathrm{y}=\mathrm{x}-5{\mathrm{x}}^2$

3. $4\mathrm{y}=2\mathrm{x}-5{\mathrm{x}}^2$

4. $\mathrm{y}=2\mathrm{x}-25{\mathrm{x}}^2$

A ship A is moving westwards with a speed of 10 km ${\mathrm{h}}^{-1}$ and a ship B, 100 km south of A is moving northwards with a speed of 10 km ${\mathrm{h}}^{-1}$. The time after which the distance between them becomes the shortest, is:

1. 5 hr

2. $5\sqrt{2}$ hr

3. $10\sqrt{2}$ 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 $\overrightarrow{\mathrm{g}}=-10 \hat{j} m{s}^{-2}$. Two particles are simultaneously projected with velocity $\overrightarrow{{\mathrm{u}}_1}=\left(10 m{s}^{-1}\right)\hat{i}+\left(10 m{s}^{-1}\right)\hat{j}$ and $\overrightarrow{{\mathrm{u}}_2}=\left(20 m{s}^{-1}\right)\hat{i}+\left(10 m{s}^{-1}\right)\hat{j}$. 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