2D Motion as Two 1D Motions

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 has initial velocity as $u_x$ i + $u_y$j and acceleration as -$a_y$. Find displacement in x when the displacement in Y becomes 0?

1. $\frac{u_x{u}_y}{a}$

2. $\frac{u_x}{a}$

3. $\frac{2u_x{u}_y}{a}$

4. $noneofthese$

Q.

The coordinates of a particle moving in a plane are given by x(t) = a cos(pt) and y(t) = a sin (pt)

where a, and p are positive contants of appropriate dimensions. Then,

1. the equation of trajectory is $x^2+y^2=a^2$

2. the velocity and acceleration of the particle are normal to each other

3. the acceleration of the particle is always directed towards the center

4. the distance travelled by the particle in time interval $(t=0tot=\frac{\pi }{2\rho }$) is "a”.

Q. Three point particles A, B and C are projected from same point with same speed at t = 0 as shown in figure.
For this situation select correct statement(s).

1. All of them reach the ground at same time.
2. All of them have same horizontal displacement when they reach the ground.
3. All of them reach the ground with same speed.
4. None

Q. This section contains 1 Matrix Match type question, which has 2 Columns (Column I and Column II). Column I has four entries (A), (B), (C) and (D), Column II has four entries (P), (Q), (R) and (S). Match the entries in Column I with the entries in Column II. Each entry in Column I may match with one or more entries in Column II.
इस खण्ड में 1 मैट्रिक्स मिलान प्रकार का प्रश्न है, जिसमें 2 कॉलम (कॉलम I तथा कॉलम II) हैं। कॉलम I में चार प्रविष्टियाँ (A), (B), (C) तथा (D) हैं, कॉलम II में चार प्रविष्टियाँ (P), (Q), (R) तथा (S) हैं। कॉलम I में दी गयी प्रविष्टियों का मिलान कॉलम II में दी गयी प्रविष्टियों के साथ कीजिए। कॉलम I में दी गयी प्रत्येक प्रविष्टि का मिलान कॉलम II में दी गयी एक या अधिक प्रविष्टियों के साथ हो सकता है।

Dimensions of physical quantity mentioned in Column-I are given in Column-II. Match the Columns and choose the appropriate answer.

कॉलम-I में प्रदर्शित भौतिक राशि की विमाऐं कॉलम-II में दी गई हैं। कॉलमों का मिलान करें और सही उत्तर चुनें

	Column I कॉलम I		Column II कॉलम II
(A)	Pressure दाब	(P)	[M0L1T–1]
(B)	Gravitational constant गुरुत्वीय नियतांक	(Q)	[M1L–1T–2]
(C)	Strain विकृति	(R)	[M–1L3T–2]
(D)	Velocity वेग	(S)	[M0L0T0]

1. A-Q, B-R, C-S, D-P
2. A-PQ, B-RS, C-S, D-P
3. A-PQ, B-R, C-RS, D-PQ
4. A-QR, B-RS, C-S, D-PQ

Q. Three point particles A, B and C are projected from same point with same speed at t = 0 as shown in figure.
For this situation select correct statement(s).

1. All of them reach the ground at same time.
2. All of them have same horizontal displacement when they reach the ground.
3. All of them reach the ground with same speed.
4. None

Q. For a 2D motion in xy plane, the motion along the x-axis is

1. always dependent on the motion along the y-axis.
2. sometimes dependent on the motion along the y-axis.
3. always independent of the motion along the y-axis.
4. none of these.

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 particle has initial velocity as $u_x$ i + $u_y$j and acceleration as -$a_y$. Find the maximum Y that it reaches?

1. $\frac{U_y}{\sqrt{2}{a}_y}$

2. $\frac{U_y^2}{\sqrt{2}{a}_y}$

3. $\frac{U_y}{\sqrt{2}{a}_y}$

4. $noneofthese$

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

Q.

A particle has initial velocity as $u_x$ i + $u_y$j and acceleration as -$a_y$. Find the maximum Y that it reaches?

1. $\frac{U_y}{\sqrt{2}{a}_y}$

2. $\frac{U_y^2}{\sqrt{2}{a}_y}$

3. $\frac{U_y}{\sqrt{2}{a}_y}$

4. $noneofthese$

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. ${30}^{\circ }$
2. ${45}^{\circ }$
3. ${50.2}^{\circ }$
4. ${60}^{\circ }$