Range on an Incline

Q. Particle A is released from a point P on a smooth inclined plane inclined at an angle $\alpha$ with the horizontal. At the same instant another particle B is projected with initial velocity $u$ making an angle $\beta$ with the horizontal. Both the particles meet again on the inclined plane. Find the relation between $\alpha$ and $\beta$

1. $\alpha +\beta ={45}^{\circ }$
2. $\alpha +\beta ={30}^{\circ }$
3. $\alpha +\beta ={90}^{\circ }$
4. $\alpha -\beta ={30}^{\circ }$

Q. A particle is launched from a horizontal plane with speed $u$ and angle of projection $\theta$. The angular velocity of the particle as observed from point of projection at the time of landing will be:

1. $\frac{g}{2u\cos \theta }$
2. $\frac{g}{u\cos \theta }$
3. $\frac{3g}{2u\cos \theta }$
4. $\frac{2g}{u\cos \theta }$

Q. Time taken by the projectile to reach from $A$ to $B$ is $t$. Then, the distance $AB$ is equal to

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

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 distance travelled by the particle along the plane before it hits the plane.

1. $30\text{m}$
2. $60\text{m}$
3. $90\text{m}$
4. $150\text{m}$

Q. If $R_1$ and $R_2$ are the maximum ranges of a projectile when it is projected from the bottom and top respectively of an inclined plane respectively, then find the maximum range of the projectile when it is projected from the ground with same velocity.

1. $\frac{R_1+R_2}{2}$
2. $\frac{R_1{R}_2}{R_1+R_2}$
3. $\frac{2R_1{R}_2}{R_1+R_2}$
4. $\frac{R_1+R_2}{R_1{R}_2}$

Q. If $R_1$ and $R_2$ are the maximum ranges of a projectile when it is projected from the bottom and top respectively of an inclined plane respectively, then find the maximum range of the projectile when it is projected from the ground with same velocity.

1. $\frac{R_1+R_2}{2}$
2. $\frac{R_1{R}_2}{R_1+R_2}$
3. $\frac{2R_1{R}_2}{R_1+R_2}$
4. $\frac{R_1+R_2}{R_1{R}_2}$

Q. A plane surface is inclined making an angle $\theta$ with the horizontal. From the bottom of this inclined plane, a bullet is fired with velocity $v$. The maximum possible range of the bullet on the inclined plane is

1. $\frac{v^2}{g}$
2. $\frac{v^2}{g(1+\sin \theta )}$
3. $\frac{v^2}{g(1-\sin \theta )}$
4. $\frac{v^2}{g(1+\cos \theta )}$

Q. A particle is projected from the bottom of an inclined plane of angle ${30}^{\circ }$ with a velocity of $20\text{m/s}$ at an angle of ${45}^{\circ }$ with the horizontal. Find the horizontal distance travelled by the particle before it hits the plane.

1. $x=\frac{80}{\sqrt{3}}[\sqrt{3}-1]\text{m}$
2. $x=\frac{40}{\sqrt{3}}[\sqrt{3}-1]\text{m}$
3. $x=\frac{40}{3}[\sqrt{3}-1]\text{m}$
4. $x=\frac{80}{3}[\sqrt{3}-1]\text{m}$

Q. Time taken by the projectile to reach from $A$ to $B$ is $t$. Then, the distance $AB$ is equal to

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

Q. Which of the following statements are true regarding a projectile projected from the bottom of the inclined plane ?

1. The direction of projection to get maximum range divides the angle between the vertical and the inclined plane in the ratio of $1:2$.
2. The direction of projection to get maximum range divides the angle between the vertical and the inclined plane in the ratio of $2:1$.
3. The direction of projection to get maximum range divides the angle between the vertical and the inclined plane in the ratio of $1:1$.
4. The direction of projection to get maximum range divides the angle between the vertical and the inclined plane in the ratio of $3:1$.

Q. A plane is inclined at an angle $\alpha ={30}^{\circ }$ with respect to the horizontal. A particle is projected with a speed $u=2\text{m/s}$ from the base of the plane, making an angle $\theta ={30}^{\circ }$ with respect to the plane as shown in the figure. The distance from the base (along the inclined plane) at which the particle hits the plane is close to

1. $27\text{cm}$
2. $20\text{cm}$
3. $18\text{cm}$
4. $14\text{cm}$

Q. A plane is inclined at an angle $\alpha ={30}^{\circ }$ with respect to the horizontal. A particle is projected with a speed $u=2\text{m/s}$, from the base of the plane, making an angle $\theta ={15}^{\circ }$ with respect to the plane as shown in the figure. The distance from the base, at which the particle hits the plane is close to [Take $g=10\text{m}/s^2]$

1. $26cm$
2. $20cm$
3. $18cm$
4. $14cm$

Q. A plane is inclined at an angle $\alpha ={30}^{\circ }$ with respect to the horizontal. A particle is projected with a speed $u=2\text{m/s}$ from the base of the plane, making an angle $\theta ={30}^{\circ }$ with respect to the plane as shown in the figure. The distance from the base (along the inclined plane) at which the particle hits the plane is close to

1. $27\text{cm}$
2. $20\text{cm}$
3. $18\text{cm}$
4. $14\text{cm}$

Q. A ball is projected down the incline plane with velocity $u$ at right angle to the plane which is inclined at an angle $\alpha$ with the horizontal. The distance $x$ along the inclined plane that it will travel before again striking the slope is

1. $\frac{2u^2}{g}\cos \alpha$
2. $\frac{2u^2}{g}\tan \alpha$
3. $\frac{2u^2}{g}\frac{\tan \alpha }{\cos \alpha }$
4. $\frac{2u^2}{g}\frac{\tan \alpha }{\sin \alpha }$

Q. Particle A is released from a point P on a smooth inclined plane inclined at an angle $\alpha$ with the horizontal. At the same instant another particle B is projected with initial velocity $u$ making an angle $\beta$ with the horizontal. Both the particles meet again on the inclined plane. Find the relation between $\alpha$ and $\beta$

1. $\alpha +\beta ={45}^{\circ }$
2. $\alpha +\beta ={30}^{\circ }$
3. $\alpha +\beta ={90}^{\circ }$
4. $\alpha -\beta ={30}^{\circ }$

Q. A particle is projected from the bottom of an inclined plane of angle ${30}^{\circ }$ with a velocity of $20\text{m/s}$ at an angle of ${45}^{\circ }$ with the horizontal. Find the horizontal distance travelled by the particle before it hits the plane.

1. $x=\frac{80}{\sqrt{3}}[\sqrt{3}-1]\text{m}$
2. $x=\frac{40}{\sqrt{3}}[\sqrt{3}-1]\text{m}$
3. $x=\frac{40}{3}[\sqrt{3}-1]\text{m}$
4. $x=\frac{80}{3}[\sqrt{3}-1]\text{m}$