The Equation for the Path of Projectile

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 equations of motion of a projectile are given by $x=36tm$ and $2y=96t-9.8t^{2}m$. The angle of projection with the horizontal 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.

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

1. Parallel to each other

2. Perpendicular to each other

3. Antiparallel to each other

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

Q. The position of a projectile launched from the origin at $t=0$ is given by $\overrightarrow{r}=(40\hat{i}+50\hat{j})\text{m}$ at $t=2\text{s}$. If the projectile was launched at an angle $\theta$ from the horizontal, then $\theta$ is
$(\text{Take}g=10{\text{ms}}^{-2})$

1. ${\tan }^{-1}\frac{2}{3}$
2. ${\tan }^{-1}\frac{3}{2}$
3. ${\tan }^{-1}\frac{7}{4}$
4. ${\tan }^{-1}\frac{4}{5}$

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

1. $y=2x-5x^2$
2. $y=x-5x^2$
3. $y=2x-1.25x^2$
4. $y=2x-25x^2$

Q. The equation of trajectory of a projectile is $y=(\frac{x}{\sqrt{3}}-\frac{g{x}^2}{20})$, where $x$ and $y$ are in meter. The maximum range of the projectile is (take $g=10m/s^2$)

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

Q. An opaque sphere of radius R lies on a horizontal plane. On the perpendicular through the point of contact there is a point source of light a distance R above the sphere then area of shadow on the plane is

1. $2\pi R^2$
2. $27\pi R^2$
3. $\pi R^2$
4. $3\pi R^2$

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

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

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

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

Q.

A ball is thrown vertically upwards with a velocity of $30m{s}^{–1}$ What will be the height reached by the body at the end of $4s$ (consider $g$ $10m{s}^{–2}$)?

1. $7.5m$

2. $3m$

3. $120m$

4. $40m$

Q.

The path of a particle moving under the influence of a force fixed in magnitude and direction is

1. Straight line

2. Circle

3. Parabola

4. Ellipse

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

Q.

A ball is projected from a point on the floor with a speed of 15 m/s at an angle of ${60}^2$ with the horizontal. Will it hit a vertical wall 5 m away from the point of projection and perpendicular to the plane of project ionm without hitting the floor ? Will the answer differ if the wall is 22 m away ?

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 acceleration due to gravity on the earth's surface at the poles is $g$ and angular veocity of the earth about the axis passing through the pole is $\omega$. An object is weight at the equator and at a height $h$ above the poles by using a spring balance. If the weight are found to be same, then $h$ is
($h<<R$, where $R$, where $R$ is the radius of the earth)

1. $\frac{R^2{\omega }^2}{2g}$
2. $\frac{R^2{\omega }^2}{g}$
3. $\frac{R^2{\omega }^2}{4g}$
4. $\frac{R^2{\omega }^2}{8g}$

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

Q. An electron enters the region between the plates of a parallel-plate capacitor with same initial velocity at an angle $\theta$ to the plants. The plate width is l and the plate separation is d. The electron follows the path shown, just missing the upper plate. Neglect gravity. Then,

1. $tan\theta =\frac{2d}{l}$
2. $tan\theta =\frac{4d}{l}$
3. $tan\theta =\frac{6d}{l}$
4. $tan\theta =\frac{8d}{l}$

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

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

Q. The particles are separated at a horizontal distance $x$ as shown in the figure. They are projected at the same time as shown in the figure with different initial speeds. The time after which the horizontal distance between the particles become zero is

1. $\frac{u}{2x}$
2. $\frac{x}{u}$
3. $\frac{2u}{x}$
4. $\frac{u}{x}$

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.

From the top of a tower of height 10 m, one fire is shot horizontally with a speed of 5*root of 3/s. Another fire is shot upwards at an angle of 60 degrees with the horizontal at some interval of time with the same speed of 5*root of 3 m/s. The shots collide in air at a certain point. The time interval between the 2 shots?

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. A ball is dropped from the top of a $100\text{m}$ high tower on a planet. In the last $\frac{1}{2}\text{s}$ before hitting the ground, it covers a distance of $19\text{m}$. Acceleration due to gravity (in ${\text{ms}}^{-2}$) near the surface on that planet is

Q. A projectile is given an initial velocity of $(\hat{i}+2\hat{j})\text{m/s}$. The cartesian equation of its path is $(g=10{\text{m/s}}^2)$

1. $y=2x-5x^2$
2. $y=x-5x^2$
3. $y=2x-1.25x^2$
4. $y=2x-25x^2$

Q.

A girl sees through a circular glass slab(refractive index 1.50 of thickness 20 mm and diameter 60 cm to the bottom of a swimming pol. Refractive index of water is 1.33. The bottom surface of the slab is in contact with the water surface.

The depth of swimming pool is 6m. The area of bottom of swimming pool that can be seen through the slab is approximately.

1. 100 m${}^2$

2. 160 m${}^2$

3. 190 m${}^2$

4. 220 m${}^2$

Q.

A particle moves along parabolic path y=2x-x² ​​​​​+2 in such a way that the x component of velocity vector remains constant(5m/s). Find the magnitude of acceleration of the particle.

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