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The postion vector of a particle is 
$\mathrm{r}=\left(\mathrm{a} \cos \mathrm{\omega t}\right)\hat{\mathrm{i}}+\left(\mathrm{a} \sin \mathrm{\omega t}\right)\hat{\mathrm{j}}$
The velocity vector of the particle is
1. parallel to position vector
2. perpendicular to position vector
3. directed towards the origin
4. directed away from the origin

A force $\mathrm{F}=\left(2\hat{\mathrm{i}}+4\hat{\mathrm{j}}\right)$ N displaces the body by $\mathrm{s}=\left(3\hat{\mathrm{j}}+5\hat{\mathrm{k}}\right)$ m in 2 s. Power generated will be
1. 11 W
2. 6 W
3. 22 W
4. 12 W

A particle moves in the x-y plane under the influence of a force such that its linear momentum is
$\mathrm{p}\left(\mathrm{t}\right)=\mathrm{A}\left[\hat{\mathrm{i}} \cos \left(\mathrm{kt}\right)-\hat{\mathrm{j}} \sin \left(\mathrm{kt}\right) \right]$
where A and k are constants. The angle between the force and momentum is
1. 0$°$
2. 30$°$
3. 45$°$
4. 90$°$

At what angle must the two forces (x+y) and (x-y) act so that the resultant may be $\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}$ ?
1. ${\cos }^{-1}\left[-\frac{{\mathrm{x}}^2+{\mathrm{y}}^2}{2\left({\mathrm{x}}^2-{\mathrm{y}}^2\right)}\right]$
2. ${\cos }^{-1}\left[-\frac{2\left({\mathrm{x}}^2-{\mathrm{y}}^2\right)}{{\mathrm{x}}^2+{\mathrm{y}}^2}\right]$
3. ${\cos }^{-1}\left[-\frac{\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)}{\left({\mathrm{x}}^2-{\mathrm{y}}^2\right)}\right]$
4. ${\cos }^{-1}\left[-\frac{\left({\mathrm{x}}^2-{\mathrm{y}}^2\right)}{\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)}\right]$

A particle is moving eastwards with a velocity of 5 ms-1. In 10 s the velocity changes to 5 ms-1 northwards. The average acceleration in this time is
1. $\frac{1}{\sqrt{2}} m{s}^{-2}$  towards north-east
2. $\frac{1}{2} {\mathrm{ms}}^{-2}$
3. zero
4. $\frac{1}{\sqrt{2}} m{s}^{-2}$  towards north-west

Consider a collection of large number of particles each with speed v. The direction of velocity is randomly distributed in the collection. The magnitude of the relative velocity between a pair of particles averaged over all the pairs in the collection is 
1. $\frac{4\mathrm{v}}{\mathrm{\pi }}$
2. greater than $\frac{4\mathrm{v}}{\mathrm{\pi }}$
3. less than $\frac{4\mathrm{v}}{\mathrm{\pi }}$
4. zero

A man stqanding on a road has to  hold his umbrella at 30^o with the vertical to keep the rain away. He throws the umbrella and starts running at 10 ${\mathrm{kmh}}^{-1}$. He finds that raindrops are hitting his head vertically. The actual speed of raindrops is 
1. 20 ${\mathrm{kmh}}^{-1}$
2. $10\sqrt{3} {\mathrm{kmh}}^{-1}$
3. $20\sqrt{3} {\mathrm{kmh}}^{-1}$
4. 10 ${\mathrm{kmh}}^{-1}$

A particels of mass m is moving in a horizontal circle of radius r, under a centripetal force $\mathrm{F}=\frac{\mathrm{k}}{{\mathrm{r}}^2}$, where k is a constant
1. The potential energy of the particle is zero
2. The potential energy of the particle is $\frac{\mathrm{k}}{\mathrm{r}}$
3. The total energy of the particle is $-\frac{\mathrm{k}}{2\mathrm{r}}$
4.  The kinetic energy of the particle is $-\frac{\mathrm{k}}{\mathrm{r}}$

A particle is moving with a constant speed v in  a circle. What is the magnitude of average after half rotation
1. 2v
2. $2\frac{\mathrm{v}}{\mathrm{\pi }}$
3. $\frac{\mathrm{v}}{2}$
4. $\frac{\mathrm{v}}{2\mathrm{\pi }}$

A point P moves in counter-clockwise direction on a circular path as shown in the figure. The movement of P is such that it sweeps out  a length $\mathrm{s}={\mathrm{t}}^3+$5 , where s is in metre and t is in second. The radius of the path is 20 m. The acceleration of P when t  = 2s is nearly
1. 13 ${\mathrm{ms}}^{-2}$
2.  12 ${\mathrm{ms}}^{-2}$
3. 7.2 ${\mathrm{ms}}^{-2}$
4. 14 ${\mathrm{ms}}^{-2}$

A can filled with water is revolved in a vertical circle of radius 4 m and the water does not fall down. The time period for a revolution is about
1. 2 s
2. 4 s
3. 8 s
4. 10 s

A particle is moving in a circle of radius R in such a way that at any instant the normal and tangential components of its acceleration are equal. If its speed at t = 0 is v₀, the time taken to complete the first revolution
1. $\frac{\mathrm{R}}{{\mathrm{v}}_0}$
2. $\frac{\mathrm{R}}{{\mathrm{v}}_0}\left(1-e^{-2\mathrm{\pi }}\right)$
3. $\frac{\mathrm{R}}{{\mathrm{v}}_0}{e}^{-2\mathrm{\pi }}$
4. $\frac{2\mathrm{\pi R}}{{\mathrm{v}}_0}$

A particle of mass m is rotating in a horizontal circle of radius R and is attached to a hanging mass M as shown in the figure. The speed of rotation required by the mass m keep M steady is 

1. $\sqrt{\frac{\mathrm{mgR}}{\mathrm{M}}}$
2. $\sqrt{\frac{\mathrm{MgR}}{m}}$
3. $\sqrt{\frac{\mathrm{mg}}{\mathrm{MR}}}$
4. $\sqrt{\frac{\mathrm{mR}}{\mathrm{Mg}}}$

A particle originally at a rest at the highest point of a smooth circle in a vertical plane, is gently pushed and starts sliding along the circle. It will leave the circle at a vertical distance h below the highest point such that

1. h = 2R
2. $\mathrm{h}=\frac{\mathrm{R}}{2}$
3. h = R
4. $\mathrm{h}=\frac{\mathrm{R}}{3}$

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

An aircraft is flying at a height of 3400 m above the ground. If the angle subtended at a ground observation point by the aircraft positions  10 s apart is 30$°$, then the speed of aircraft is 
1. 19.63 ${\mathrm{ms}}^{-1}$
2. 1963 ${\mathrm{ms}}^{-1}$
3. 108 ${\mathrm{ms}}^{-1}$
4. 196.3 ${\mathrm{ms}}^{-1}$

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 a projectile about the point of projection when the particle is at its maximum height H is
1. $\frac{\sqrt{3}}{2}\frac{{\mathrm{mv}}^2}{\mathrm{g}}$
2. zero
3. $\frac{{\mathrm{mv}}^3}{\sqrt{2}\mathrm{g}}$
4. $\frac{\sqrt{3}}{16}\frac{{\mathrm{mv}}^3}{\mathrm{g}}$

Two bodies are projected from the same point with equal speeds in search directions that they both strike the same points on a plane whose inclination is $\mathrm{\beta }$. If $\mathrm{\alpha }$ be the angle of projection of the first body with the horizontal the ratio of their times of light is
1. $\frac{\cos \mathrm{\alpha }}{\sin \left(\mathrm{\alpha }+\mathrm{\beta }\right)}$
2. $\frac{\sin \left(\mathrm{\alpha }+\mathrm{\beta }\right)}{\cos \mathrm{\alpha }}$
3. $\frac{\cos \mathrm{\alpha }}{\sin \left(\mathrm{\alpha }-\mathrm{\beta }\right)}$
4. $\frac{\sin \left(\mathrm{\alpha }-\mathrm{\beta }\right)}{\cos \mathrm{\alpha }}$

Three balls are dropped from the top fo a building with equal speed at different angles. When the balls strike ground their velocities 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 moves along a parabolic path $\mathrm{y}=9{\mathrm{x}}^2$ in such a way that the x- components of velocity remains constant and has a value $\frac{1}{3}{\mathrm{ms}}^{-1}$. The acceleration of the projectile is 
1. $\frac{1}{3}\hat{\mathrm{j}} {\mathrm{ms}}^{-2}$
2. $3\hat{\mathrm{j}} {\mathrm{ms}}^{-2}$
3. $\frac{2}{3}\hat{\mathrm{j}} {\mathrm{ms}}^{-2}$
4. $2 \hat{\mathrm{j}} {\mathrm{ms}}^{-2}$

If $\overrightarrow{\mathrm{A}}=\overrightarrow{\mathrm{B}}+\overrightarrow{\mathrm{C}}$ and the magnitudes of $\overrightarrow{\mathrm{A}},\overrightarrow{\mathrm{B}},\overrightarrow{\mathrm{C}}$ are 5,4 and 3 units, then angle between $\overrightarrow{\mathrm{A} }\mathrm{and} \overrightarrow{\mathrm{C}}$ is
1. ${\cos }^{-1}\left(\frac{3}{5}\right)$
2. ${\cos }^{-1}\left(\frac{4}{5}\right)$
3. ${\sin }^{-1}\left(\frac{3}{4}\right)$
4. $\frac{\mathrm{\pi }}{2}$

A particle moves in the xy plane with only an x-component of acceleration of 2 ${\mathrm{ms}}^{-2}$. The particle starts from the origin at t = 0 with an initial velocity having an x-component of 8 ${\mathrm{ms}}^{-1}$ and y-component of -15 ${\mathrm{ms}}^{-1}$. The total velocity vector at any time t is
1. $\left[\left(8+2\mathrm{t}\right)\hat{\mathrm{i}}-15\hat{\mathrm{j}}\right]{\mathrm{ms}}^{-1}$
2. zero
3. $2\mathrm{t}\hat{\mathrm{i}}+15\hat{\mathrm{j}}$
4. directed along z-axis

A unit vector along incident ray of lights is $\hat{\mathrm{i}}$. The unit vector for the corresponding refracted ray of lights 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. $\hat{\mathrm{i}}\times \hat{\mathrm{n}}=\mathrm{\mu }\left(\hat{\mathrm{n}}+\overrightarrow{\mathrm{r}}\right)$
2. $\hat{\mathrm{i}}.\hat{\mathrm{n}}=\mathrm{\mu }\left(\hat{\mathrm{n}}.\hat{\mathrm{r}}\right)$
3. $\hat{\mathrm{i}}\times \hat{\mathrm{n}}=\mathrm{\mu }\left(\hat{r}\times \hat{\mathrm{n}}\right)$
4. $\mathrm{\mu }\left(\hat{\mathrm{i}}\times \hat{\mathrm{n}}\right)=\hat{\mathrm{r}}\times \hat{\mathrm{n}}$

In the figure, the time taken by the projectile to reach from A to B is t. Then the distance AB is equal to

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. $\frac{\sqrt{3}}{2}$
3. 0.5
4. 1 

A particle is projected with a certain velocity at an angle $\mathrm{\alpha }$ above the horizontal from the foot of an inclined plane of inclination $30°$. If the particle strikes the plane normally then $\mathrm{\alpha }$ is equal to 
1. $30°+{\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right)$
2. 45$°$
3. 60$°$
4. $30°+{\tan }^{-1}\left(2\sqrt{3}\right)$

The trajectory of a particle in a vertical plane is $\mathrm{y}=\mathrm{ax}-{\mathrm{bx}}^2$, where a and b are constants and x and y are respective 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
1. $\frac{{\mathrm{b}}^2}{2\mathrm{a}\text{'}}{\tan }^{-1}\left(\mathrm{b}\right)$
2. $\frac{a^2}{\mathrm{b}\text{'}}{\tan }^{-1}\left(2\mathrm{b}\right)$
3. $\frac{a^2}{4\mathrm{b}\text{'}}{\tan }^{-1}\left(a\right)$
4. $\frac{\mathit{2}{a}^2}{\mathrm{b}\text{'}}{\tan }^{-1}\left(a\right)$

A particle is projected from the ground with an initial speed of v at an angle $\mathrm{\theta }$ with horizontal. The average velocity of the particle between its point of projection and highest point of trajectory is 
1. $\frac{\mathrm{v}}{2}\sqrt{1+2{\cos }^2\mathrm{\theta }}$
2. $\frac{\mathrm{v}}{2}\sqrt{1+2{\sin }^2\mathrm{\theta }}$
3. $\frac{\mathrm{v}}{2}\sqrt{1+3{\cos }^2\mathrm{\theta }}$
4. $\mathrm{vcos} \mathrm{\theta }$

Two balls A and B are thrown with speeds u and u/2, respectively. Both the balls cover the same horizontal distance before returning to the plane of projection. If the angle of projection of ball B is 15^o with the horizontal, then the angle of projection of A is 
1. ${\sin }^{-1}\left(\frac{1}{8}\right)$
2. $\frac{1}{2}{\sin }^{-1}\left(\frac{1}{8}\right)$
3. $\frac{1}{3}{\sin }^{-1}\left(\frac{1}{8}\right)$
4. $\frac{1}{4}{\sin }^{-1}\left(\frac{1}{8}\right)$