Time of Flight with Incline as Frame of Reference

Q. The maximum range of a bullet fired from a toy pistol mounted on a car at rest is $R_0=40\text{m}$. What will be the acute angle of inclination of the pistol for maximum range when the car is moving in the direction of firing with uniform velocity $v=20\text{m/s}$ on a horizontal surface ?
$($ Take $g=10{\text{m/s}}^2)$

1. ${60}^{\circ }$
2. ${30}^{\circ }$
3. ${75}^{\circ }$
4. ${45}^{\circ }$

Q. A particle is projected with a certain velocity at an angle $\alpha$ above the horizontal from the foot of an inclined plane of inclination ${30}^{\circ }$. If the particle strikes the plane normally. then $\alpha$ is equal to

1. ${60}^{\circ }$
2. ${30}^{\circ }+{\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right)$
3. ${30}^{\circ }+{\tan }^{-1}\left(\frac{1}{2\sqrt{3}}\right)$
4. ${45}^{\circ }$

Q. A ball is projected from an inclined plane having inclination $\theta$, normal to the plane with speed $u$. It strikes the vertical wall horizontally and again it stikes the inclined plane. If all the collisions are elastic, find the time taken by the ball to hit the inclined plane.

1. $T=\frac{2u\sin \theta }{g}$
2. $T=\frac{2u\cos \theta }{g}$
3. $T=\frac{2u\tan \theta }{g}$
4. $T=\frac{u\sin \theta }{g}$

Q. In figure, a crate slides down an inclined right angled trough. Acceleration of the crate is

1. $2.1{\text{m/s}}^2$
2. $2.5{\text{m/s}}^2$
3. $2.7{\text{m/s}}^2$
4. $2.9{\text{m/s}}^2$

Q. A particle is projected from the bottom of an inclined plane of angle ${30}^{\circ }$ with a velocity of $30\text{m/s}$ at an angle of ${60}^{\circ }$ with the horizontal. Find the time of flight.
(Take $g=10m/s^2$)

1. $\sqrt{3}\text{s}$
2. $\frac{1}{\sqrt{3}}\text{s}$
3. $2\sqrt{3}\text{s}$
4. $2\text{s}$

Q.

A copper rod of length L and mass m is sliding down a smooth inclined plane of inclination $\theta$ with a constant
speed v. A current I is flowing in perpendicular to the plane of diagram inwards. A vertically upward magnetic
field $\overrightarrow{B}$ exist in this region.The magnitude of the required magnetic field $\overrightarrow{B}$ is

1. $\frac{mg}{IL}sin\theta$

2. $\frac{mg}{IL}sin\theta$

3. $\frac{mg}{IL}tan\theta$

4. $\frac{mg}{ILsin\theta }$

Q.

A bullet fired at an angle of $30°$ with the horizontal hits the ground $3.0km$ away. By adjusting its angle of projection, can one hope to hit a target $5.0km$ away? Assume the muzzle speed to be fixed, and neglect air resistance.

Q. A ball is thrown upward at an angle of ${30}^{\circ }$ with the horizontal and lands on the top edge of a building that is $20m$ away. The top edge is $5m$ above the throwing point. The initial speed of the ball in $metre/second$ is (take g = 10 $m/s^2$)

1. $15m{s}^{-1}$
2. $20m{s}^{-1}$
3. $25m{s}^{-1}$
4. $30m{s}^{-1}$

Q.

A ball A is projected from O with an initial velocity $v_0=7m{s}^{-1}$ in a direction $37°$ above the horizontal. Another ball B, $3m$ from O on a line $37°$ above the horizontal is released from rest at the instant A starts, as shown in figure.

[Take $\sin 37°=\frac{3}{7};\cos 37°=\frac{4}{5};g=9.8m{s}^{-2}$]

Based on above information, answer the following questions:

How far will B have fallen when it is hit by A?

1. $\frac{1}{10}m$

2. $\frac{9}{10}m$

3. $\frac{196}{25}m$

4. $\frac{81}{10}m$

Q. A glass ball collides with a smooth horizontal surface (xz plane) with a velocity $V=ai-bj$. If the coefficient of restitution of collision be e, the velocity of the ball just after the collision will be :

1. $\sqrt{e^2{a}^2+b^2}$ at angle $ta{n}^{-1}\left(\frac{a}{eb}\right)$ to the vertical
2. $\sqrt{a^2+e^2{b}^2}$ at angle $ta{n}^{-1}\left(\frac{a}{eb}\right)$ to the vertical
3. $\sqrt{a^2+\frac{b^2}{e^2}}$ at angle $ta{n}^{-1}\left(\frac{ea}{b}\right)$ to the vertical
4. $\sqrt{\frac{a^2}{e^2}+b^2}$ at angle $ta{n}^{-1}\left(\frac{a}{eb}\right)$ to the vertical

Q. A particle is projected at angle ${37}^o$ with the incline plane in upward direction with speed $10$m/s. The angle of incline plane is given ${53}^o$. Then the maximum height attained by the particle from the incline plane will be:

1. $3m$
2. $4m$
3. $5m$
4. zero

Q. A ball is projected from the ground with speed $u$ at an angle $\alpha$ with horizontal. It collides with a wall at distance $a$ from the point of projection and returns to its original position. Find the coefficient of restitution between the ball and the wall.

1. $\frac{1}{(\frac{u^2\sin 2\alpha }{ag}-1)}$
   
2. $\frac{1}{(\frac{u^2\sin 2\alpha }{2ag}-1)}$
3. $\frac{1}{(\frac{u\sin 2\alpha }{ag}-1)}$
   
4. 
   
   $\frac{1}{(\frac{u^2\sin 2\alpha }{g}-1)}$
   

Q. A particle is projected with a certain velocity at an angle $\alpha$ above the horizontal from the foot of an inclined plane of inclination ${30}^{\circ }$. If the particle strikes the plane normally. then $\alpha$ is equal to

1. ${30}^{\circ }+{\tan }^{-1}\left(\frac{1}{2\sqrt{3}}\right)$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. ${30}^{\circ }+{\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right)$

Q. A man standing on a road hold his umbrella at ${30}^0$ with the vertical to keep the rain away. He throws the umbrella and starts running at 10 km / hr. He finds that raindrops are hitting his head vertically, the speed of raindrops with respects to the road will be -

1. 40 km / hr
2. 10 km / hr
3. 20 km / hr
4. 30 km / hr

Q. A particle is projected from the bottom of an inclined plane of angle ${30}^{\circ }$ with a velocity of $30\text{m/s}$ at an angle of ${60}^{\circ }$ with the horizontal. Find the time of flight.
(Take $g=10m/s^2$)

1. $\sqrt{3}\text{s}$
2. $\frac{1}{\sqrt{3}}\text{s}$
3. $2\sqrt{3}\text{s}$
4. $2\text{s}$

Q. A body is projected from the ground at an angle of ${60}^o$ to the horizontal with a speed $20m/s$. the radius of the curvature of the path of the particle when its velocity makes an angle of ${30}^o$ to the horizontal.

1. $12.6m$
2. $14.3m$
3. $17.3m$
4. $15.4m$

Q. A body is projected at an angle of ${30}^{\circ }$ with the horizontal and with a speed of $30m{s}^{-1}$. What is the angle with the horizontal after $1.5s$? $(g=10m{s}^{-2})$.

1. ${60}^{\circ }$
2. $0^{\circ }$
3. ${90}^{\circ }$
4. ${30}^{\circ }$

Q. Find time of flight of a projectile thrown horizontally with speed $10m{s}^{-1}$ from a long inclined plane which makes an angle of $\theta ={45}^o$ with the horizontal (Take $g=10m{s}^{-2}$)

1. $\sqrt{2}sec$
2. $2\sqrt{2}sec$
3. $2sec$
4. $none$

Q. A particle is projected with a certain velocity at an angle $\alpha$ above the horizontal from the foot of an inclined plane of inclination ${30}^{\circ }$. If the particle strikes the plane normally. then $\alpha$ is equal to

1. ${30}^{\circ }+{\tan }^{-1}\left(\frac{1}{2\sqrt{3}}\right)$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. ${30}^{\circ }+{\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right)$

Q.

Q. A particle is projected up with a velocity of $v_0$ = 10 m/s at an angle pf ${\theta }_0={60}^0$ with horizontal onto an inclined plane. The angle of inclination of the plane is ${30}^0$. The time of flight of the particle till it strikes the plane is (take g = 10 $m/s^2$)

Q. If collision between the block and the incline is completely elastic, then the vertical (upward) component of the velocity of the block at point $B$, immediately after it strikes the second incline is

1. $\sqrt{30}m/s$
2. $\sqrt{15}m/s$
3. $0$
4. $-\sqrt{15}m/s$

Q. A body is projected horizontally from the top of a tower with initial velocity $18m{s}^{-1}$. It hits the ground at angle ${45}^o$. What is the vertical component of velocity when it strikes the ground?

Q. Three projectiles $A$, $B$ and $C$ are projected at an angle of ${30}^o$, ${45}^o$, ${60}^o$ respectively. If $R_A$, $R_B$ and $R_C$ are ranges of $A$, $B$ and $C$ respectively, then (velocity of projection is same for $A$, $B$ and $C$):

1. $R_A=R_B=R_C$
2. $R_A=R_C>R_B$
3. $R_A<R_B<R_C$
4. $R_A=R_C<R_B$

Q. When a man moves down the inclined plane with a constant speed $5m{s}^{-1}$ which makes an angle of ${37}^o$ with the horizontal, he finds that the rain is falling vertically downward. When he moves up the same inclined plane with the same speed, he finds that the rain makes an angle $\theta =ta{n}^{-1}\left(\frac{7}{8}\right)$ with the horizontal. The speed of the rain is

1. $\sqrt{32}m{s}^{-1}$
2. $\sqrt{116}m{s}^{-1}$
3. $5m{s}^{-1}$
4. $\sqrt{73}m{s}^{-1}$

Q. Two fixed frictionless inclined plane making an angle ${30}^o$ and ${60}^o$ with the vertical are shown in the figure. Two blocks A and $B$ are placed on the two planes. What is the relative vertical acceleration of A with respect to $B$ ?

1. $4.9m{s}^{-2}$ in horizontal direction
2. $4.9m{s}^{-2}$ in vertical direction
3. $9.8m{s}^{-2}$ in vertical direction
4. zero

Q. Single Correct Answer Type
A particle is projected from the ground with an initial speed of 'v' at angle $\theta$ with horizontal. the average velocity of the particle between its point of projection and height point of trajectory is

1. $\frac{v}{2}\sqrt{1+2{\cos }^2\theta }$
2. $\frac{v}{2}\sqrt{1+{\cos }^2\theta }$
3. $v\cos \theta$
4. $\frac{v}{2}\sqrt{1+3{\cos }^2\theta }$

Q. A particle is moving in a plane with velocity given by $y={\nu }_{0}i+\left(\alpha \omega cos\omega t\right)j,$where i , j are units vectors along $x$ and $y$axes respectively .The trajectory of the particle if the particle starts from origin at $t=0$ will be

1. $y=tanx$
2. $y=\alpha cos\frac{\omega x}{{\nu }_0}$
3. $y=\alpha sin\frac{\omega x}{{\nu }_0}$
4. $y=cotx$

Q. A ball strikes a wall with a velocity $\overrightarrow{u}$ at an angle $\theta$ with the normal to the wall surface and rebounds from it at an angle $\beta$ with the surface. Then :

1. $(\theta +\beta )<{90}^o,$ if the wall is smooth
2. if the wall is rough, coefficient of restitution $<\frac{tan\beta }{cos\theta }$
3. if the wall is rough, coefficient of restitution $=\frac{tan\beta }{cos\theta }$
4. none of the above

Q. If R is the range on a horizontal plane and T the time of flight of a projectile, then the angle of projection $\alpha$ given by $\tan \alpha$ is equal to?

1. $g{T}^2/2R$
2. $gT/2R$
3. $T^2/Rg$
4. $T^2/2Rg$