2D Motion as Two 1D Motions

Q. A ball is projected upwards from the top of tower with a velocity 50 m/s making an angle ${30}^{\circ }$ with the horizontal. The height of tower is 70 m. After how many seconds from the instant of throwing will the ball reach the ground

1. 2 s
2. 5 s
3. 7 s
4. 9 s

Q. The speed of a projectile when it is at its maximum height is $\sqrt{\frac{2}{5}}$ times its speed at half the maximum height. The angle of projection is

1. ${60}^{\circ }$
2. ${45}^{\circ }$
3. ${\tan }^{-1}\frac{3}{4}$
4. ${30}^{\circ }$

Q.

A particle moves in a straight line in such a way that its velocity at any point is given by $v^2=2–3x$, where $x$ is measured from a fixed point. The acceleration is

1. Uniform

2. Zero

3. Non-uniform

4. None

Q.

How gravitational force is conservative force!???

Q. From the top of a $19.6\text{m}$ high tower, a ball is thrown horizontally. If the line joining the point of projection to the point where it hits the ground, makes an angle of ${45}^{\circ }$ with the horizontal, then the initial velocity of the ball is

1. $9.8\text{m/s}$
2. $4.9\text{m/s}$
3. $14.7\text{m/s}$
4. $2.8\text{m/s}$

Q. A stone is projected from the ground with velocity 25 m/s . Two seconds later, it just clears a wall 5 m high. The angle of projection of the stone is (g=10 m/$s^2$)

1. 30^o
2. 45^o
3. 50.2^o
4. 60^o

Q. A particle of mass $m$ is projected with a speed of $u$ from the ground at an angle $\theta =\frac{\pi }{3}$ w.r.t. horizontal $\left(\text{x-axis}\right)$. When it has reached its maximum height, it collides completely inelastically with another particle of the same mass and velocity $u\hat{i}$. The horizontal distance covered by the combined mass before reaching the ground is:

1. $\frac{3\sqrt{3}}{8}\frac{u^2}{g}$
2. $\frac{3\sqrt{2}}{4}\frac{u^2}{g}$
3. $\frac{5}{8}\frac{u^2}{g}$
4. $2\sqrt{2}\frac{u^2}{g}$

Q. A projectile has initially the same horizontal velocity as it would acquire if it had moved from rest with uniform acceleration of $3{\text{m/s}}^2$ for $0.5$ min. If the maximum height reached by it is $80\text{m}$, then the angle of projection is ($g=10{\text{m/s}}^2$)

1. ${\tan }^{-1}3$
2. ${\tan }^{-1}\frac{3}{2}$
3. ${\tan }^{-1}\frac{4}{9}$
4. ${\sin }^{-1}\frac{4}{9}$

Q. A particle is projected from the bottom of an inclined plane of inclination ${30}^{\circ }$ (with the horizontal) with speed $40\text{m/s}$ at an angle ${60}^{\circ }$ with the horizontal. The speed of the particle when its velocity vector is parallel to inclined plane is $(g=10{\text{m/s}}^2)$

1. $\frac{40}{\sqrt{3}}\text{m/s}$
2. $20\text{m/s}$
3. $20\sqrt{3}\text{m/s}$
4. $10\text{m/s}$

Q.

A particle leaves the origin with an initial velocity $\overrightarrow{v}$ = ( $3.00\hat{i}$ ) m/s and a constant acceleration $\overrightarrow{a}$ = ( $-1.00\hat{i}-0.500\hat{j}$ ) m/$s^2$. When it reaches its maximum x coordinate, match its position vector and velocity vector(Coloumn A) with vectors (Coloumn B)?

Coloumn A	Coloumn B
(i) Position vector	(U) $-10\hat{j}$
(ii) Velocity vector	(V) $4.5\hat{i}-22.5\hat{j}$
	(W) $4.5\hat{i}-21.5\hat{j}$
	(X) $-15\hat{j}$

1. (i) - U, (ii) - V

2. (i) - X, (ii) - W

3. (i) - V, (ii) - X

4. (i) - W, (ii) - X

Q. Find range of projectile on the inclined plane which is projected perpendicular to the incline plane with velocity $20\text{m/s}$ as shown in the figure.

Q.

The distance x covered in time t by a body having velocity $v_0$ and having a constant acceleration a is given by $x=v_{0}t+\frac{1}{2}a{t}^2$. This result follows from

1. Newtons first law

2. Newtons second law

3. Newtons third law

4. none of these

Q. A particle moves in the $x-y$ plane and at time $t$, it is at the point whose coordinates are $(t^2,t^3-2t)$. Then at what instant of time will its velocity and acceleration vectors be perpendicular to each other?

1. $\frac{1}{3}\text{sec}$
2. $\frac{3}{2}\text{sec}$
3. $\frac{2}{3}\text{sec}$
4. $\text{It will never happen}$

Q. The radius vector of a point A relative to the origin varies as $\overrightarrow{r}=at\hat{i}+b{t}^2\hat{j}$ where a and b are positive constants. Find the equation of trajectory.

1. $y=\frac{b}{a^2}{x}^2$
2. $y^2=\frac{b}{a^2}x$
3. $y=\frac{a^2}{b}{x}^2$
4. $\text{none of these}$

Q. A ball is projected on smooth inclined plane in direction perpendicular to line of greatest slope with velocity of $8\text{m/s}$ .Find its speed after $1\text{sec}$ (take $g=10{\text{m/s}}^2$ ).

1. $8\text{m/s}$
2. $12\text{m/s}$
3. $10\text{m/s}$
4. $14\text{m/s}$

Q. A liquid stands at the plane level in U-tube when at rest. If areas of cross-section of both the limbs are equal, what will be the difference in heights $h$ of the liquid in the two limbs of U-tube, when the system is given an acceleration $a$ in horizontal direction towards right as shown ?

1. $\frac{g}{a}\frac{L^2}{H}$
2. $\frac{La}{g}$
3. $\frac{L^2}{H}\frac{a}{g}$
4. $\frac{Lg}{a}$

Q. Two particles $\text{A}$ and $\text{B}$ are projected simultaneously from two towers of height $10\text{m}$ and $20\text{m}$ respectively. Particle $\text{A}$ is projected with an initial speed of $10\sqrt{2}\text{m/s}$ at an angle of ${45}^{\circ }$ with horizontal, while particle $\text{B}$ is projected horizontally with speed $10\text{m/s}$. If they collide in air, what is the distance $d$ between the towers and time taken for collision?
$(g=10{\text{m/s}}^2)$

1. $20\text{m},1\text{s}$
2. $18\text{m},3\text{s}$
3. $16\text{m},8\text{s}$
4. $15\text{m},2\text{s}$

Q. A body falling freely from a given height H hits an inclined plane in its path at a height h. As a result of this impact, the direction of the velocity of the body becomes horizontal. For what value of $\left(\frac{h}{H}\right)$ the body will take maximum time to reach the ground?

Q.

A point moves in x-y plane according to the law x=4 sin 6t and y=4(1-cos 6t). The distance traversed by the particle in 4 seconds is (x and y are in meters)

1. 96 m

2. 48 m

3. 24 m

4. 108 m

Q. A racing car has uniform acceleration of $4$ $m/s^2$. What distance will it cover in $10$ seconds after the start?

1. $100$m
2. $300$m
3. $200$m
4. $400$m

Q. A particle moves in $xy-$plane according to the equation $x=\alpha t$ and $y=\alpha t(1-\beta t)$ where $\alpha$ and $\beta$ are positive constants and $t$ is time.

Choose the correct option in regard to the trajectory of the particle.

1. Equation of the trajectory of the particle is $y=x(1+\frac{\beta }{\alpha }x)$
2. Equation of the trajectory of the particle is $y=x(1+\frac{\alpha }{\beta }x)$
3. Trajectory of the particle is symmetric about $x=\frac{2\alpha }{\beta }$
4. Trajectory of the particle is symmetric about $x=\frac{\alpha }{2\beta }$

Q.

A particle moves in a straight line so that its velocity at any point is given $v^2=a+bx$, where $a$, $b\ne 0$ are constants. The acceleration is

1. Zero

2. Uniform

3. Non-uniform

4. Intermediate

Q. Why gun recoils when bullet is been fired?
Is it because of conservation of momentum or this happens that's why it called conservation of momentum?

Q. If the position vector of a point $P$ relative to the origin varies with time as, $\overrightarrow{r}=2\alpha t\hat{i}-\beta t^2\hat{j}$ , where $\alpha$ and $\beta$ are constants, then the equation for trajectory of the particle is

1. $y=\frac{-\beta x^2}{2{\alpha }^2}$
2. $y=\frac{-\beta x^2}{2\alpha }$
3. $y=\frac{-\beta x^2}{4\alpha }$
4. $y=\frac{-\beta x^2}{4{\alpha }^2}$

Q. A ball is projected from a point on an inclined plane horizontally. It takes $\frac{1}{\sqrt{2}}$ seconds to strike the plane. If the incline plane makes an angle of ${30}^{\circ }$ with horizontal, then find the range of the ball on incline plane in meters. [g = 10 $m/s^2$] ___

Q. Motion of the tip of second hand of the clock is an example of

1. one dimensional motion.
2. motion in a plane with uniform velocity.
3. motion in a plane with constant acceleration.
4. none of these.

Q. A projectile is thrown with a velocity of $50{\text{ms}}^{-1}$ at an angle of ${53}^{\circ }$ with the horizontal. The equation of the trajectory is given
(Take $g=10{\text{m/s}}^2$)

1. $180y=240x-x^2$
2. $180y=x^2-240x$
3. $180y=135x-x^2$
4. $180y=x^2-135x$

Q. From the top of building 80 m high a ball is thrown horizontally which hits the ground at a distance. A line joining the top of the building to the joint where it hits the ground makes an angle ^{${45}^0$} with the ground. Find its initial velocity.

1. 10
2. 20
3. 30
4. 40

Q. A particle is moving with a uniform acceleration. Its displacement at any instant t is given by s= 10t +4.9$t^2$. What is (i) initial velocity (ii)velocity at t=3 second and (iii) the uniform acceleration?

Q. A particle is projected with a velocity $u$, at an angle $\alpha$, with the horizontal. Time at which its vertical component of velocity becomes half of its net speed at the highest point will be

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