The Equation for the Path of Projectile

Q. The path of projectile is represented by $y=Px-Q{x}^2$
$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{(a)}& \text{Range}& \text{(p)}& \frac{P}{Q}\\ \text{(b)}& \text{Maximum height}& \text{(q)}& P\\ \text{(c)}& \text{Time of flight}& \text{(r)}& \frac{P^2}{4Q}\\ \text{(d)}& \text{Tangent of angle of projection}& \text{(s)}& \sqrt{\frac{2}{Qg}}P\end{array}$

1. $a\to s,b\to q,c\to r,d\to q$
2. $a\to p,b\to r,c\to q,d\to q$
3. $a\to p,b\to r,c\to s,d\to q$
4. $a\to p,b\to q,c\to r,d\to s$

Q. A heavy particle is projected with a velocity at an angle with the horizontal into a uniform gravitational field. The slope of the trajectory of the particle varies not according to which of the following curves?

1. 
2. 
3. 
4. 

Q. A heavy particle is projected with a velocity at an angle with the horizontal into a uniform gravitational field. The slope of the trajectory of the particle varies not according to which of the following curves?

1. 
2. 
3. 
4. 

Q.

At the top of the trajectory of a projectile, the directions of its velocity and acceleration are

1. Perpendicular to each other

2. Parallel to each other

3. Inclined to each other at an angle of ${45}^{\circ }$

4. Antiparallel to each other

Q. A projectile is given an initial velocity of $(2\hat{i}+4\hat{j})$$m/s$, where $\hat{i}$ is along the ground and $\hat{j}$ is along the vertical. If $g=10m/s^2$, then the equation of its trajectory is

1. $y=x-5x^2$
2. $y=2x-\frac{5}{4}{x}^2$
3. $y=4x-5x^2$
4. $y=4x-25x^2$

Q. The equation of projectile is $y=16x-\frac{5x^2}{4}$. The horizontal range is-

1. $16\text{m}$
2. $8\text{m}$
3. $3.2\text{m}$
4. $12.8\text{m}$

Q. The equation of projectile is $y=8x-\frac{5x^2}{2}$. The horizontal range is-

1. $8\text{m}$
2. $3.2\text{m}$
3. $8.4\text{m}$
4. $2\text{m}$

Q. A particle is projected over a triangle from one extremity of its horizontal base. Grazing over the vertex, it falls on the other extremity of the base. If $\alpha$ and $\beta$ be the base angles of the triangle and 𝜃 the angle of projection, then which of the following holds true ?

1. $tan\theta =\tan \alpha +tan\beta$
2. $tan\alpha =\tan \theta +tan\beta$
3. $tan\beta =\tan \theta +tan\alpha$​​​​​​​
4. $tan\theta =\tan \alpha -tan\beta$

Q. Trajectory of particle in a projectile motion is given as $y=x-\frac{x^2}{80}$. Here, $x$ and $y$ are in metre. For this projectile motion match the following and select proper option $(g=10{\text{m/s}}^2)$:
$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{(A)}& \text{Angle of projection}& \text{(p)}& 20\text{m}\\ \text{(B)}& \text{Angle of velocity}& \text{(q)}& 80\text{m}\\ & \text{with horizontal}& & \\ & \text{after}4s& & \\ \text{(C)}& \text{Maximum height}& \text{(r)}& {45}^{\circ }\\ \text{(D)}& \text{Horizontal range}& \text{(s)}& {\tan }^{-1}\left(\frac{1}{2}\right)\end{array}$

1. $A\to s,B\to s,C\to q,D\to p$
2. $A\to r,B\to r,C\to q,D\to p$
3. $A\to r,B\to r,C\to p,D\to q$
4. $A\to s,B\to r,C\to p,D\to q$

Q. A projectile is fired from the ground at a speed of 20 m/s at an angle of ${37}^{\circ }$ to the horizontal, then $[g=10m/s^2]$

1. Radius of curvature at firing point is 50 m.
2. As projectile moves, its radius of curvature decreases continuosly.
3. As projectile moves its radius of curvature first decreases and then it increases.
4. Minimum radius of curvature is 30 m.

Q. If a particle is projected from origin and it follows the trajectory $y=x-\frac{1}{4}{x}^2$, then the time of flight is ($g$ = acceleration due to gravity)

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

Q. The equation of motion of a projectile are given by x = 36 t metre and $2y=96t–9.8t^2$ metre. The angle of projection is

1. $si{n}^{-1}\left(\frac{4}{5}\right)$
2. $si{n}^{-1}\left(\frac{3}{5}\right)$
3. $si{n}^{-1}\left(\frac{4}{3}\right)$
4. $si{n}^{-1}\left(\frac{3}{4}\right)$

Q. A projectile is projected from the level ground in vertical x-y plane (vertical is y axis). The point of projection is considered as the origin. For which of the following initial velocities of projection u and angle of projection $\theta$ with the horizontal will the projectile pass through the point (4, 2) (all in m) (Neglect air friction)

1. $u=4\sqrt{5}\frac{m}{s},\theta ={45}^{\circ }$
2. $u=4\sqrt{5}\frac{m}{s},\theta =ta{n}^{-1}\left(3\right)$
3. $u=2\sqrt{5}\frac{m}{s},\theta ={45}^{\circ }$
4. $u=8\sqrt{5}\frac{m}{s},\theta =ta{n}^{-1}\left(3\right)$

Q. At the top of the trajectory of a projectile, the acceleration is

1. Maximum
2. Minimum
3. Zero
4. g

Q. The trajectory of a projectile is given by the equation $y=x-x^2$. What is the value of the initial velocity and the angle of projection?

1. $\sqrt{10}\text{m/s},{30}^{\circ }$
2. $10\text{m/s},{45}^{\circ }$
3. $\sqrt{10}\text{m/s},{45}^{\circ }$
4. $10\text{m/s},{30}^{\circ }$

Q. A projectile is projected from the level ground in vertical x-y plane (vertical is y axis). The point of projection is considered as the origin. For which of the following initial velocities of projection u and angle of projection $\theta$ with the horizontal will the projectile pass through the point (4, 2) (all in m) (Neglect air friction)

1. $u=4\sqrt{5}\frac{m}{s},\theta ={45}^{\circ }$
2. $u=4\sqrt{5}\frac{m}{s},\theta =ta{n}^{-1}\left(3\right)$
3. $u=2\sqrt{5}\frac{m}{s},\theta ={45}^{\circ }$
4. $u=8\sqrt{5}\frac{m}{s},\theta =ta{n}^{-1}\left(3\right)$

Q. The trajectory of a projectile is given by the equation $y=x-x^2$. What is the value of the initial velocity and the angle of projection?

1. $\sqrt{10}\text{m/s},{30}^{\circ }$
2. $10\text{m/s},{45}^{\circ }$
3. $\sqrt{10}\text{m/s},{45}^{\circ }$
4. $10\text{m/s},{30}^{\circ }$

Q. An object is projected with a velocity of $20\text{m/s}$ 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 has value of $(g=10m{s}^{-2})$

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

Q. The height $y$ and the distance $x$ along the horizontal plane of a projectile on a certain planet (assuming flat surface) with no surrounding atmosphere are given by $x=6t$ and $y=8t-5t^2$, where $x$ and $y$ are in metre and time $t$ is in second. Find the velocity in $m{s}^{-1}$ with which the body is projected. Take acceleration due to gravity $g=10{\text{m/s}}^2$

Q. A stone is thrown from a point at a distance $a$ from a wall of height $b$. If it just clears the wall and angle of projection is $\alpha$, the maximum height attained by the stone is

1. $\frac{a^2{\sec }^2\alpha }{4(a\sec \alpha -b)}$
2. $\frac{a^2{\tan }^2\alpha }{4(a\tan \alpha -b)}$
3. $\frac{a^2\tan \alpha }{4(a-b\sec \alpha )}$
4. $\frac{a^2\tan \alpha }{4}$

Q. If a particle is projected from origin and it follows the trajectory $y=x-\frac{1}{2}{x}^2$, then the time of flight is ($g$ = acceleration due to gravity)

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

Q. A projectile is fired from the ground at a speed of 20 m/s at an angle of ${37}^{\circ }$ to the horizontal, then $[g=10m/s^2]$

1. Radius of curvature at firing point is 50 m.
2. As projectile moves, its radius of curvature decreases continuosly.
3. As projectile moves its radius of curvature first decreases and then it increases.
4. Minimum radius of curvature is 30 m.

Q. If a particle is projected from origin and it follows the trajectory $y=x-\frac{1}{2}{x}^2$, then the time of flight is ($g$ = acceleration due to gravity)

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

Q. A projectile is given an initial velocity of $(2\hat{i}+4\hat{j})$$m/s$, where $\hat{i}$ is along the ground and $\hat{j}$ is along the vertical. If $g=10m/s^2$, then the equation of its trajectory is

1. $y=x-5x^2$
2. $y=2x-\frac{5}{4}{x}^2$
3. $y=4x-5x^2$
4. $y=4x-25x^2$

Q. A gun is fired horizontally on the bull's eye at a height $h$. Then

1. the bullet hits the bull's eye
2. the bullet moves left or right of the Bull's eye due to jerk experienced on firing
3. the bullet misses the bull's eye and hits upward
4. the bullet misses the target and hits downwards