Range on an Incline

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

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

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

Q. Explain how $u\cos \theta$ is the horizontal component and $u\sin \theta$ the vertical component when a projectile is projected at an angle $\theta$ from the horizontal.

Q. An electric field is acting vertically upwards. A small body of mass 1 gm and charge –1$\mu C$ is projected with a velocity 10 m/s at an angle ${45}^{\circ }$ with horizontal. Its horizontal range is 2m then the intensity of electric field is: (g = 10 $m/s^2$)

1. 20, 000 N/C
2. 10, 000 N/C
3. 40, 000 N/C
4. 90, 000 N/C

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

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

Q. Two balls are thrown from an inclined plane at angle of projection $\alpha$ with the plane one up the incline plane and other down the incline as shown in the figure. If $R_1$ & $R_2$ be their respective ranges and $h_1$ & $h_2$ be their respective maximum height then
[here $T_1$ & $T_2$ are times of flight in the two cases respectively]

1. $h_1=h_2$
2. $R_2-R_1=T_1^2$
3. $R_2-R_1=\text{g sin}\theta T_2^2$
4. $R_2-R_1=\text{g sin}\theta T_1^2$

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

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 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. A ball is projected horizontally with a speed $v$ from the top of a plane inclined at an angle ${45}^{\circ }$ with the horizontal. How far from the point of projection will the ball strike the plane?

1. $\sqrt{2}\left[\frac{2v^2}{g}\right]$
2. $\frac{v^2}{g}$
3. $\sqrt{2}\frac{v^2}{g}$
4. $\frac{2v^2}{g}$

Q. An inclined plane makes as angle ${\theta }_0={30}^{\circ }$ with the horizontal. A particle is projected from this plane with a speed of $5\text{m/s}$ at an angle of elevation $\beta ={30}^{\circ }$ with the horizontal as shown in the figure.


Find the range of the particle on the plane when it strikes the plane. (Assume the incline is long enough and $g=10{\text{m/s}}^2$)

1. $3.5\text{m}$
2. $4\text{m}$
3. $4.2\text{m}$
4. $5\text{m}$

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. 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. The path of a projectile is given by $y=ax-b{x}^2$. What is the value of the angle ${\theta }_0$ and speed $v_0$ , about a point at which the projectile is launched?

1. ${\theta }_0={\cos }^{-1}\frac{1}{\sqrt{b^2+1}},v_0=\sqrt{\frac{g}{2b}(1+a^2)}$
2. ${\theta }_0={\cos }^{-1}\frac{1}{\sqrt{a^2-1}},v_0=\sqrt{\frac{g}{2b}(1+a^2)}$
3. ${\theta }_0={\cos }^{-1}\frac{1}{\sqrt{a^2+1}},v_0=\sqrt{\frac{g}{2b}(1+a^2)}$
4. ${\theta }_0={\cos }^{-1}\frac{1}{\sqrt{a^2+1}},v_0=\sqrt{\frac{g}{2a}(1+b^2)}$

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. 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. 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 ball is projected with a speed $v_0$ at an angle of $37°$ with the horizontal, from the top of a cliff as shown in the figure. The ball has been thrown with the aim so that it just hits directly the edge of the cliff, but the ball just misses the target and falls on ground as shown in figure. [Take $\sin 37°=\frac{3}{5};\cos 37°=\frac{4}{5};g=10m{s}^{-2}$.]

Based on above information, answer the following question:
The value of $v_0$ is

1. $25m{s}^{-1}$

2. $30m{s}^{-1}$

3. $45m{s}^{-1}$

4. $18m{s}^{-1}$

Q.

A particle is projected at an angle of ${37}^{\circ }$ with an inclined plane. The inclined plane is at an angle of ${60}^{\circ }$ with the horizontal. Find
I. Time of flight of particle.
II. Distance traveled by particle (AB) along the inclined plane

1. I. Time of flight $=4.8s$
   II. Distance traveled $=11.4m$

2. I. Time of flight $=2.4s$
   II. Distance traveled $=5.7m$

3. I. Time of flight $=1.2s$
   II. Distance traveled $=5.7m$

4. I. Time of flight $=6.4s$
   II. Distance traveled $=11.4m$

Q.

If $R$ and $H$ represent horizontal range and maximum height of the projectile, then the angle of projection with the horizontal is

1. ${\tan }^{-1}\left(\frac{H}{R}\right)$

2. ${\tan }^{-1}\left(\frac{2H}{R}\right)$

3. ${\tan }^{-1}\left(\frac{4H}{R}\right)$

4. ${\tan }^{-1}\left(\frac{4R}{H}\right)$

Q. A particle is projected with some velocity $u$ making an angle ${60}^0$ with horizontal up an inclined plane. The plane makes an angle ${30}^0$ with horizontal. The particle's range on the inclined plane with it's time of flight $T$, is :

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

Q. A pendulum of mass $m$ hangs from a support fixed to a trolley. The direction of the string (ie angle $\theta$) when the trolley rolls up a plane of inclination $\alpha$ with acceleration $a$ is :

1. zero
2. ${\tan }^{-1}\alpha$
3. ${\tan }^{-1}\frac{\alpha +g\sin \alpha }{g\cos \alpha }$
4. ${\tan }^{-1}\frac{a}{g}$

Q. Two balls are thrown from an inclined plane at angle of projection $\alpha$ with the plane one up the incline plane and other down the incline as shown in the figure. If $R_1$ & $R_2$ be their respective ranges and $h_1$ & $h_2$ be their respective maximum height then
[here $T_1$ & $T_2$ are times of flight in the two cases respectively]

1. $h_1=h_2$
2. $R_2-R_1=T_1^2$
3. $R_2-R_1=\text{g sin}\theta T_2^2$
4. $R_2-R_1=\text{g sin}\theta T_1^2$

Q. A body is projected with velocity $24m{s}^{-1}$ making an angle ${30}^{\circ }$ with the horizontal. The vertical component of its velocity after $2s$ is $(g=10m{s}^{-2})$

1. $8m{s}^{-1}$ upward
2. $-8m{s}^{-1}$ downward
3. $32m{s}^{-1}$ upward
4. $32m{s}^{-1}$ downward

Q. A man with some passengers in his boat, starts perpendicular to the flow of river $200m$ wide and flowing with $2m/s$. Speed of boat in still water is $4m/s$. When he reaches half the width of river, the passengers asked him that they want to reach the point just opposite to the end from where they have started. If the direction due which he must row to reach the required end is $\alpha =2ta{n}^{-1}\left(\frac{1}{x}\right)$. Find $x$.

Q. Two masses $M_1$ and $M_2$ are attached to the ends of a light string, which passes over a massless pulley, attached to the top of a double inclined smooth plane of angle s, of inclination $\alpha$ and $\beta$. If $M_2$>$M_1$ and $\beta$>$alpha$, then the acceleration of block $M_2$ down the inclined will be ?

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. If $p_1,p_2,p_3$ are respectively the perpendiculars from the vertices of a triangle to the opposite sides, then $p_1{p}_2{p}_3=$

1. $\frac{a^2{b}^2{c}^2}{R^2}$
2. $\frac{a^2{b}^2{c}^2}{{4R}^2}$
3. $\frac{4a^2{b}^2{c}^2}{R^2}$
4. $\frac{a^2{b}^2{c}^2}{{8R}^2}$

Q. Find the proper length of a rod if in the laboratory frame of reference its velocity is $v=\frac{c}{2}$, the length $l=1.00m$, and the angle between the rod and its direction of motion is $\theta ={45}^{\circ }$.