Projectile Time, Height and Range

Q. When an object is shot from the bottom of a long smooth inclined plane kept at an angle $60°$with horizontal, it can travel a distance $x_1$ along the plane. But, when the inclination is decreased to $30°$ and the same object is shot with the same velocity, it can travel $x_2$ distance. Then $x_1:x_2$ will be

1. $1:\sqrt{2}$
2. $\sqrt{2}:1$
3. $1:\sqrt{3}$
4. $1:2\sqrt{3}$

Q. A particle is projected with a velocity $v$ so that its range on a horizontal plane is twice the greatest height attained. Consider $g=10{\text{m/s}}^2$ as acceleration due to gravity. Then, its range will be:

1. $\frac{4v^2}{5g}$
2. $\frac{4v^2}{5g^2}$
3. $\frac{4v}{5g^2}$
4. $\frac{4v^3}{5g^2}$

Q. A particle is projected at an angle of ${60}^{\circ }$ above the horizontal with a speed of $10m/s$. After some time, the direction of its velocity makes an angle of ${30}^{\circ }$ above the horizontal. The speed of the particle at this instant is

1. $\frac{5}{\sqrt{3}}m/s$
2. $5\sqrt{3}m/s$
3. $5m/s$
4. $\frac{10}{\sqrt{3}}m/s$

Q.

A projectile is fired from the surface of the earth with a velocity of $5{\text{ms}}^{-1}$ and angle $\theta$ with the horizontal. Another projectile fired from another planet with a velocity of $3{\text{ms}}^{-1}$ at the same angle follows a trajectory which is identical with the trajectory of the projectile fired from the earth. The value of the acceleration due to gravity on the planet (in ${\text{ms}}^{-2}$) is

(Given $g=9.8{\text{ms}}^{-2}$)

1. $3.5$
2. $5.9$
3. $16.3$
4. $110.8$

Q. The trajectory of a projectile in a vertical plane is given by $y=ax-b{x}^2,$ where $a$ and $b$ are constants and $x$ and $y$ are respectively 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. $\frac{b^2}{2a},{\tan }^{-1}\left(b\right)$
2. $\frac{a^2}{b},{\tan }^{-1}\left(2a\right)$
3. $\frac{a^2}{4b},{\tan }^{-1}\left(a\right)$
4. $\frac{2a^2}{b},{\tan }^{-1}\left(a\right)$

Q. A metallic rod of mass per unit length $0.5kg{m}^{-1}$ is lying horizontally on a smooth inclined plane which makes an angle of ${30}^o$ with the horizontal. The rod is not allowed to slide down by a passing a current through it when a magnetic field of induction $0.25T$ is acting on it in the vertical direction. The current flowing in the rod to keep it stationary is

1. $14.76A$
2. 
   $7.14A$
3. 
   $11.32A$
4. 
   $5.98A$

Q. The maximum range of a gun on a horizontal terrain is $16\text{km}$. If $g=10{\text{m/s}}^2$, then muzzle velocity of the shell must be

1. $160\text{m/s}$
2. $200\sqrt{2}\text{m/s}$
3. $400\text{m/s}$
4. $800\text{m/s}$

Q. The horizontal range and maximum height attained by a projectile are $R$ and $H$, respectively. If a constant horizontal acceleration $a=\frac{g}{4}$ is imparted to the projectile due to wind, then its horizontal range and maximum height will be

1. $(R+H),\frac{H}{2}$
2. $(R+H),2H$
3. $(R+2H),H$
4. $(R+H),H$

Q. At what angle with the horizontal should a ball be thrown so that the range $R$ is related to the time of flight $\left(T\right)$ as $R=5T^2?(\text{Take g}=10{\text{m/s}}^2)$

1. ${30}^{\circ }$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. ${90}^{\circ }$

Q. A train is moving along a straight line with a constant acceleration. A boy standing inside the train throws a ball in forward direction with a speed of $10\text{m/s}$ at an angle of ${60}^{\circ }$ to the horizontal, with respect to himself. The boy has to move forward by $1.15\text{m}$ inside the train to catch the ball (at the initial height). Then,
[Take $g=10{\text{m/s}}^2$]

1. Acceleration of train is $5{\text{m/s}}^2$
2. Acceleration of ball is $10{\text{m/s}}^2$
3. Acceleration of ball is $5\sqrt{5}{\text{m/s}}^2$
4. Displacement of train is $8.66\text{m}$

Q. A ball is projected from the ground at an angled of ${45}^o$ with the horizontal surface. It reaches a maximum height of $120m$ and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic energy. Immediately after the bounce, the velocity of the ball makes an angle of ${30}^o$ with the horizontal surface. The maximum height it reaches after the bounce, (in metres), is

Q. The position coordinates of a projectile projected from ground on a certain planet (with no atmosphere) are given by $y=(4t-2t^2)\text{m}$ and $x=\left(3t\right)\text{m}$, where $t$ is in $\text{second}$ and point of projection is taken as origin. The angle of projection of projectile with vertical is

1. ${30}^{\circ }$
2. ${37}^{\circ }$
3. ${45}^{\circ }$
4. ${60}^{\circ }$

Q. An object is projected with a velocity of 20 m/s making an angle of ${45}^{\circ }$ with horizontal. The equation for the trajectory is h = Ax – B$x^2$ where h is height, x is horizontal distance, A and B are constants. The ratio A : B is (g = 10 m$s^{-2}$)

1. 1 : 5
2. 1 : 40
3. 40 : 1
4. 5 : 1

Q. For a projectile thrown with a speed $v$, the horizontal range is $\frac{\sqrt{3}{v}^2}{2g}$. The vertical range is $\frac{v^2}{8g}$. The angle which the projectile makes with the horizontal initially is

1. ${15}^{\circ }$
2. ${30}^{\circ }$
3. ${45}^{\circ }$
4. ${60}^{\circ }$

Q. A particle is projected from the ground with velocity $u$ at an angle $\theta$ with horizontal. The horizontal range, maximum height and time of flight are $R,H$ and $T$ respectively. They are given by, $R=\frac{u^2\sin 2\theta }{g},H=\frac{u^2{\sin }^2\theta }{2g}\text{and}T=\frac{2u\sin \theta }{g}$
Now keeping $u$ as fixed, $\theta$ is varied from ${30}^{\circ }$ to ${60}^{\circ }$. Then

1. $R$ will first increase then decrease, $H$ will increase and $T$ will decrease
2. $R$ will first increase then decrease while $H$ and $T$ both will increase
3. $R$ will decrease while $H$ and $T$ will increase
4. $R$ will increase while $H$ and $T$ will increase

Q. If $R$ and $h$ represent the horizontal range and maximum height respectively of an oblique projectile, then $\frac{R^2}{8h}+2h$ represents

1. Maximum horizontal range
2. Time of flight
3. Maximum verticle range
4. Velocity of projectile at highest point.

Q. A projectile can have the same range for two angles of projection. If $h_1$ and $h_2$ are maximum heights when the range in the two cases is $R$, then the relation between $R,h_1$ and $h_2$ is

1. $R=4\sqrt{h_1{h}_2}$
2. $R=2\sqrt{h_1{h}_2}$
3. $R=\sqrt{h_1{h}_2}$
4. None of these

Q. A particle starts from the origin of co-ordinates at time $t=0$ and moves in the $xy$ plane with a constant acceleration $\alpha$ in the $y$ – direction. Its equation of motion $y=\beta x^2$. Its velocity component in the $x$-direction is

1. $\frac{\alpha }{\beta }$
2. $\frac{\sqrt{2\alpha }}{\beta }$
3. $\frac{\alpha }{2\beta }$
4. $\sqrt{\frac{\alpha }{2\beta }}$

Q.

A body of mass $1kg$ is projected from ground at an angle ${30}^{\circ }$ with horizontal on a level ground at a speed $50m/s$. The magnitude of change in momentum of the body during its flight is $(g=10m/s^2)$

1. $50kgm{s}^{-1}$
2. $100kgm{s}^{-1}$
3. $25kgm{s}^{-1}$
4. Zero

Q. At a height of $0.4m$ from the ground, the velocity of a projectile in vector form is$\overrightarrow{v}=(6\hat{i}+2\hat{j})m/s.$ The angle of projection is
$(g=10m/s^2)$

1. ${30}^{\circ }$
2. ${60}^{\circ }$
3. ${45}^{\circ }$
4. ${37}^{\circ }$

Q. Ratio of minimum kinetic energies of two projectiles of same mass is $4:1$. The ratio of the maximum height attained by them is also $4:1$. The ratio of their ranges would be

1. $16:1$
2. $4:1$
3. $8:1$
4. $2:1$

Q. The horizontal range and the maximum height of a projectile are equal. The angle of projection of the projectile is

1. ${\tan }^{-1}\left(\frac{1}{4}\right)$
2. ${\tan }^{-1}\left(2\right)$
3. $\frac{\pi }{4}$
4. ${\tan }^{-1}\left(4\right)$

Q. A particle projected from ground moves at an angle ${45}^{\circ }$ with horizontal one second after projection and speed is minimum two seconds after the projection. The angle of projection of particle is
[Neglect the effect of air resistance]

1. ${\tan }^{-1}\left(3\right)$
2. ${\tan }^{-1}\left(2\right)$
3. ${\tan }^{-1}\left(4\right)$
4. ${\tan }^{-1}\left(\sqrt{2}\right)$

Q. A particle is projected at an angle of ${30}^{\circ }$ w.r.t horizontal with speed $20m/s.$ Take the point of projection as the origin of the coordinate axes and find the velocity vector of the particle after 1 s (in $m/s$).

1. $5\sqrt{3}\hat{i}+5\hat{j}$
2. $5\hat{j}$
3. $5\hat{i}+10\sqrt{3}\hat{j}$
4. $10\sqrt{3}\hat{i}$

Q. A ball is projected vertically upwards with a certain initial speed. Another ball of the same mass is projected at an angle of ${60}^{\circ }$ with the vertical with the same initial speed. At the highest point, the ratio of their potential energies will be

1. 3:2
2. 2:3
3. 2:1
4. 4:1

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

1. $0^{\circ }$
2. ${30}^{\circ }$
3. ${45}^{\circ }$
4. ${60}^{\circ }$

Q. The velocity at the maximum height of a projectile is $\frac{\sqrt{3}}{2}$ times its initial velocity of projection $\left(u\right)$. Its range on the horizontal plane is

1. $\frac{\sqrt{3}{u}^2}{2g}$
2. $\frac{3u^2}{2g}$
3. $\frac{3u^2}{g}$
4. $\frac{u^2}{2g}$

Q. The equation of a projectile is $y=ax-b{x}^2$. Its horizontal range is

1. $\frac{a}{b}$
2. $\frac{b}{a}$
3. $a+b$
4. $b-a$

Q. A player kicks a football at an angle of ${45}^{\circ }$ with a velocity of $20m{s}^{-1}$. A second player $60m$ away along the direction of kick starts running to receive the ball at that instant. Find the speed with which second player should run to reach the ball before it hits the ground $(g=10m{s}^{-2})$

1. $5\sqrt{2}m/s$
2. $\sqrt{2}m/s$
3. $3\sqrt{2}m/s$
4. $1m/s$

Q. A particle is fired at an angle of ${30}^{\circ }$ to the horizontal such that the vertical component of its initial velocity is $80m/s$. Its time of flight is $T$. Its velocity at $t=\frac{T}{4}$ has a magnitude of nearly
(Take $g=10m/s^2$)

1. $200m/s$
2. $300m/s$
3. $144m/s$
4. $100m/s$