Please be patient as the PDF generation may take up to a minute.

A body is projected with velocity ${\mathrm{v}}_1$ from the point A as shown in fig. At the same time another body is projected vertically upwards from B with velocity ${\mathrm{v}}_2$. The point B lies vertically below the highest point. For both the bodies to collide $\frac{{\mathrm{v}}_2}{{\mathrm{v}}_1}$ should be 

1. 2
2. $\sqrt{\frac{3}{2}}$
3. 0.5
4. 1

The trajectory of a projectile in a vertical plane is $\mathrm{y}=\mathrm{ax}-{\mathrm{bx}}^2$, where a and b are constants and x and y are respectivley horizontal and vertical distances of the projectile from the point of projection. The maximum height attained by the particle and the angle of projection from the horizontal are 
1. $\frac{{\mathrm{b}}^2}{2\mathrm{a}\text{'}}{\tan }^{-1}\left(b\right)$
2. $\frac{a^2}{\mathrm{b}\text{'}}{\tan }^{-1}\left(2b\right)$
3. $\frac{a^2}{4\mathrm{b}\text{'}}{\tan }^{-1}\left(\mathrm{a}\right)$
4. $\frac{\mathit{2}{a}^2}{\mathrm{b}\text{'}}{\tan }^{-1}\left(\mathrm{a}\right)$

Two inclined places OA and OB intersect in a horizontal plane having their inclinations $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ with the horizontal as shown in figure. A particle is projected from point P with velocity u along a direction perpendicular to plane OA. The particle strikes plane OB perpendicularly at Q

If $\mathrm{\alpha }=30°, \mathrm{\beta }=30°$, the time of flight from P to Q is 
1. $\frac{\mathrm{u}}{\mathrm{g}}$
2. $\frac{\sqrt{3}\mathrm{u}}{\mathrm{g}}$
3. $\frac{\sqrt{2}\mathrm{u}}{\mathrm{g}}$
4. $\frac{2\mathrm{u}}{\mathrm{g}}$

A particle is moving with velocity $\overrightarrow{\mathrm{v}}=K\left(y\hat{i}+x\hat{j}\right)$, where K is a constant. The general equation for its path is 
1. ${\mathrm{y}}^2={\mathrm{x}}^2+\mathrm{constant}$
2. $\mathrm{y}={\mathrm{x}}^2+\mathrm{constant}$
3. ${\mathrm{y}}^2=\mathrm{x}+\mathrm{constant}$
4. $\mathrm{yx}=\mathrm{constant}$

The kinetic energy k of a particle moving along a circle of radiuss R depends on the distance covered s as $\mathrm{k}={\mathrm{as}}^2$ where a is a constant. The force acting on the particle is 
1. $2\mathrm{a}\frac{{\mathrm{s}}^2}{\mathrm{R}}$
2. $2\mathrm{as}{\left(1+\frac{{\mathrm{s}}^2}{{\mathrm{R}}^2}\right)}^{1/2}$
3. 2as
4. $2\mathrm{a}\frac{{\mathrm{R}}^2}{s}$

A stone tied to a string of length L is whirled in a vertical circle with the other end of the string at the centre. At a certain instant of time, the stone is at its lowest position and has a speed u. The magnitude of the charge in its velocity as it reaches a position where the string is horizontal is 
1. $\sqrt{{\mathrm{u}}^2-2\mathrm{gl}}$
2. $\sqrt{2\mathrm{gl}}$
3. $\sqrt{{\mathrm{u}}^2-\mathrm{gl}}$
4. $\sqrt{2\left({\mathrm{u}}^2-\mathrm{gl}\right)}$

A body is projected vertically upwards at time t=0 and it is seen at H at instants t₁ and t₂ seconds during its flight. The maximum height attained is (g is acceleration due to gravity)
1. $\frac{\mathrm{g}{\left({\mathrm{t}}_2-{\mathrm{t}}_1\right)}^2}{8}$ 
2. $\frac{\mathrm{g}{\left({\mathrm{t}}_2+{\mathrm{t}}_1\right)}^2}{4}$
3. $\frac{\mathrm{g}{\left({\mathrm{t}}_2+{\mathrm{t}}_1\right)}^2}{8}$
4. $\frac{\mathrm{g}{\left({\mathrm{t}}_2-{\mathrm{t}}_1\right)}^2}{4}$

A particle of mass m is projected with a velocity v making an angle of $30°$ with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height h is 
1. $\frac{\sqrt{3}}{2}\frac{m{v}^2}{g}$
2. zero
3. $\frac{{\mathrm{mv}}^3}{\sqrt{2}\mathrm{g}}$
4. $\frac{\sqrt{3}}{16}\frac{m{v}^3}{g}$

A unit vector along incident ray of light is $\hat{\mathrm{i}}$. The unit vector for the corresponding refracted ray of light is $\hat{\mathrm{r}}$. $\hat{\mathrm{n}}$ is a unit vector normal to the boundary of the medium and directed towards the incident medium. If m be the refractive index of the medium, then Snell's law (2nd) of refraction is 
1. $\hat{\mathrm{i}}\times \hat{n}=\mu \left(\hat{n}+\overrightarrow{r}\right)$
2. $\hat{\mathrm{i}}.\hat{n}=\mu \left(\hat{r}.\hat{n}\right)$
3. $\hat{\mathrm{i}}\times \hat{n}=\mu \left(\hat{r}\mathit{\times }\hat{n}\right)$
4. $\mu \left(\hat{\mathrm{i}}\times \hat{n}\right)=\hat{\mathrm{r}}\times \hat{n}$

Two particles \(\mathrm{A}\) and \(\mathrm{B}\), move with constant velocities \(\overrightarrow{{v}_1}\) and \(\overrightarrow{{v}_2}\) respectively. At the initial moment, their position vectors are \(\overrightarrow{{r}_1}\) and \(\overrightarrow{{r}_2}\) respectively. The condition for particles \(\mathrm{A}\) and \(\mathrm{B}\) for their collision will be:
1. \(\vec{r_1} \cdot \vec{v_1}=\vec{r_2} \cdot \vec{v_2}\)
2. ​​​​​\(\dfrac{\vec{r_1}-\vec{r_2}}{\left|\vec{r_1}-\vec{r_2}\right|}=\dfrac{\vec{v_2}-\vec{v_1}}{\left|\vec{v_2}-\vec{v_1}\right|}\)${\mathrm{}}_{}$
3. \(\vec{r_1} \times \vec{v_1}=\vec{r_2} \times \vec{v_2}\)${\mathrm{ \; \; }}_{}$
4. \(\vec{r_1}-\vec{r_2}=\vec{v_1}-\vec{v_2}\) 

Three balls are dropped from the top of a building with equal speeds at different angles. When the balls strike ground , their speeds are ${\mathrm{v}}_1,{\mathrm{v}}_2 \mathrm{and} {\mathrm{v}}_3$ respectively,then

1. ${\mathrm{v}}_1>{\mathrm{v}}_2>{\mathrm{v}}_3$
2. ${\mathrm{v}}_3>{\mathrm{v}}_2>{\mathrm{v}}_1$
3. ${\mathrm{v}}_1={\mathrm{v}}_2={\mathrm{v}}_3$
4. ${\mathrm{v}}_1<{\mathrm{v}}_2<{\mathrm{v}}_3$

A particle of mass m is projected with a velocity v making an angle of 45$°$ with the horizontal. The magnitude of the angular momentum of the particle about the point of projection when the particle is at its maximum height, is
1. $\mathrm{m}\sqrt{2{\mathrm{gh}}^3}$
2. $\frac{{\mathrm{mv}}^3}{\sqrt{2}\mathrm{g}}$
3. $\frac{{\mathrm{mv}}^3}{4\sqrt{2}\mathrm{g}}$
4. zero

A gun is mounted on the top of a tower 100 m high. At what angle should the gun be inclined to cover a maximum range of firing on the gorund below the muzzle speed of the bullet is 150 m/s (take g = 10 $m/s^2$) :
1. 43.8 $°$
2. ${\mathrm{x}}_{\mathrm{m}}=\frac{\mathrm{u}\sqrt{{\mathrm{u}}^2+2\mathrm{gh}}}{2\mathrm{g}}$
3. 53$°$
4. ${\mathrm{x}}_{\mathrm{m}}=\frac{\sqrt{{\mathrm{u}}^2+2\mathrm{gh}}}{\mathrm{g}}$

A ball is projected with a velocity ${\mathrm{V}}_0$ at an angle $\mathrm{\theta }$ from a point of projection, then :
1. Its velocity is normal to the initial velocity of projection at $\mathrm{t}=\frac{{\mathrm{v}}_0}{\mathrm{gsin\theta }} \mathrm{for} 0<\mathrm{\theta }\le \frac{\mathrm{\pi }}{2}$
2. Its velocity is normal to the initial velocity of projection at $\mathrm{t}=\frac{{\mathrm{v}}_0}{\mathrm{gsin\theta }} \mathrm{for} 0<\mathrm{\theta }\le \frac{\mathrm{\pi }}{4}$
3. Its velocity is normal to the initial velocity of projection at $\mathrm{t}=\frac{{\mathrm{v}}_0}{\mathrm{gsin\theta }} \mathrm{for} \frac{\mathrm{\pi }}{4}<\mathrm{\theta }\le \frac{\mathrm{\pi }}{2}$
4. Its velocity is normal to the initial velocity of projection at $\mathrm{t}=\frac{{\mathrm{v}}_0}{\mathrm{gsin\theta }}$ for all values of $\mathrm{\theta }$

A small particle of mass m is projected at an angle $\mathrm{\theta }$ with the x-axis with an initial velocity ${\mathrm{v}}_0$ in the x-y plane as shown in the figure. At a time $\mathrm{t}<\frac{{\mathrm{v}}_0\mathrm{sin\theta }}{\mathrm{g}}$, the angular momentum of the particle is 

1. $\frac{1}{2}mg{v}_0{t}^2\cos \theta \hat{i}$
2. $-{\mathrm{mgv}}_0{\mathrm{t}}^2\mathrm{cos\theta }\hat{\mathrm{j}}$
3. $-{\mathrm{mgv}}_0\mathrm{tcos\theta }\hat{k}$
4. $-\frac{1}{2}mg{v}_0{t}^2\cos \theta \hat{k}$