Q51
Allen
2. NAT
Numerical
A rocket is fired vertically upwards with initial velocity 40 m/s at the ground level. Its engines then fired and it is accelerated at $2 \, m/s^{2}$ until it reaches an altitude of 1000 m. At that point the engines shut off and the rocket goes into free-fall. If the velocity (in m/s) just before it collides with the ground is $40\alpha$ . Then fill the value of $\alpha$ . Disregard air resistance ( $g = 10m/s^{2}$ ).
Correct Answer: 4
Explanation:
Ans. (4) $v^{2} = u^{2} + 2$ as $v^{2} = 40^{2} + 2(2)(1000) = 5600$ $v^{\prime 2} = v^{2} + 2gs$ $\Rightarrow v^{\prime 2} = 5600 + 2\times 10\times 100$ $\Rightarrow v^{\prime} = 160 = 40\alpha$ $\Rightarrow \alpha = 4$
Q52
Allen
2. NAT
Numerical
The vertical height y and horizontal distance x of a projectile on a certain planet are given by $x = (3t)m$ , $y = (4t - 6t^{2})m$ where t is in seconds. Find the speed of projection (in m/s).
Correct Answer: 5
Explanation:
Ans. (5) $v_{x} = \frac{dx}{dt} = 3, v_{y} = \frac{dy}{dt} = 4 - 12t$ At $t = 0, \vec{u} = u_{x}\hat{i} + u_{y}\hat{j} = 3\hat{i} + 4\hat{j}$ Speed of projection $= u = \sqrt{3^2 + 4^2} = 5m / s$
Q53
Allen
2. NAT
Numerical
A particle is thrown with a speed $60 \, ms^{-1}$ at an angle $60^{\circ}$ to the horizontal. When the particle makes an angle $30^{\circ}$ with the horizontal in downward direction, it's speed at that instant is v. What is the value of $v^{2}$ in SI units?
Correct Answer: 1200
Explanation:
Ans. (1200) Horizontal velocity remain constant. $60\cos 60^{\circ} = v\cos 30^{\circ}$ $v = 20\sqrt{3}$ $v^{2} = 1200$
Q54
Allen
2. NAT
Numerical
A cricketer can throw a ball to a maximum horizontal distance of 100 m. How much high (in m) above the ground can the cricketer throw the same ball?
Correct Answer: 50
Explanation:
Ans. (50)
$H _ {\mathrm{max}} = u ^ {2} \sin^ {2} \theta / 2 g = u ^ {2} / 2 g$
$R _ {\max} = \left(u ^ {2} \sin 2 \theta / g\right) _ {\max} = u ^ {2} / g$
$2 H _ {\mathrm{max}} = \mathrm{Range}$
$2 H _ {\mathrm{max}} = 1 0 0$
$H _ {\mathrm{max}} = 5 0 m$
$H _ {\mathrm{max}} = u ^ {2} \sin^ {2} \theta / 2 g = u ^ {2} / 2 g$
$R _ {\max} = \left(u ^ {2} \sin 2 \theta / g\right) _ {\max} = u ^ {2} / g$
$2 H _ {\mathrm{max}} = \mathrm{Range}$
$2 H _ {\mathrm{max}} = 1 0 0$
$H _ {\mathrm{max}} = 5 0 m$
Q55
Allen
2. NAT
Numerical
A particle is projected upwards with a velocity of 100 m/s at an angle of $60^{\circ}$ with the vertical. Find the time (in sec) when the particle will move perpendicular to its initial direction, taking $g = 10 \, m/s^{2}$ .
Correct Answer: 20
Explanation:
Ans. (20) $U = 100\cos 30\hat{t} +100\sin 30^{\circ}\hat{j}$ $\vec{V} = 10\cos 30\hat{t} +(100\sin 30 - 10t)\hat{j}$ $\vec{U}.\vec{V} = 0$ $\Rightarrow \frac{100}{10\times\frac{1}{2}} = 20\sec$
Q56
Allen
2. NAT
Numerical
A ball is thrown horizontally from a cliff such that it strikes ground after 5 s. The line of sight from the point of projection to the point of landing makes an angle of $37^{\circ}$ with the horizontal. If the initial velocity of projection is $\frac{50x}{3}$ . Find x.
Correct Answer: 2
Explanation:
Ans. (2) $t^2 = \frac{2h}{g} = 25$ $\Rightarrow h = 125m$ $x = u_{x}\times t$ $h\cot 37^{\circ} = u_{x}\times 5$ $u_{x} = 100 / 3$
Q57
Allen
2. NAT
Numerical
A ball is dropped from rest from a tower of height 5m. As a result of the wind it lands at a distance 6m from the bottom of the tower as shown. Assuming no air resistance but that the wind gives the ball a constant horizontal velocity v. Find value of v in m/s.
Correct Answer: 6
Explanation:
Ans. (6)
$\begin{array}{l} {x = \sqrt {\frac {2 h}{g}} v} \\ {\Rightarrow \quad v = \frac {x}{\sqrt {\frac {2 h}{g}}} = \frac {6}{\sqrt {\frac {2 \times 5}{1 0}}} = 6 m / s} \end{array}$
$\begin{array}{l} {x = \sqrt {\frac {2 h}{g}} v} \\ {\Rightarrow \quad v = \frac {x}{\sqrt {\frac {2 h}{g}}} = \frac {6}{\sqrt {\frac {2 \times 5}{1 0}}} = 6 m / s} \end{array}$
Q58
Allen
2. NAT
Numerical
A cuboidal elevator cabin is shown in the figure. A ball is thrown from point A on the floor of cabin when the elevator is falling under gravity. The plane of motion is ABCD and the angle of projection of the ball with AB, relative to elevator, if the ball collides with point C, is $\alpha$ . Then find the value of tan $\alpha$ .
Correct Answer: 5
Explanation:
Ans. (5)
$\begin{array}{l} \tan \alpha = \frac {A D}{A B} \\ \frac {2 . 5}{0 . 5} \\ \tan \alpha = 5 \end{array}$
$\begin{array}{l} \tan \alpha = \frac {A D}{A B} \\ \frac {2 . 5}{0 . 5} \\ \tan \alpha = 5 \end{array}$
Q59
Allen
2. NAT
Numerical
Two particles P and Q are launched simultaneously as shown in figure. Find the minimum distance between particles in meters.
Correct Answer: 6
Explanation:
Ans. (6)
$\begin{array}{r l} & {\vec {v} _ {P Q} = \vec {v} _ {P} - \vec {v} _ {Q}} \\ & {\qquad = \left(\frac {1 0}{\sqrt {2}} \hat {\imath} + \frac {1 0}{\sqrt {2}} \hat {j}\right) - \left(- \frac {7 0}{\sqrt {2}} \hat {\imath} + \frac {7 0}{\sqrt {2}} \hat {j}\right)} \\ & {\qquad = 4 0 \sqrt {2} \hat {\imath} - 3 0 \sqrt {2} \hat {j}} \\ & {A B = 9 0 \times \frac {4}{3} = 1 2 0 m} \\ & {\Rightarrow \quad Q B = 1 0 m} \\ & {\Rightarrow \quad Q M = 1 0 \sin 3 7 ^ {\circ} = 1 0 \times \frac {3}{5} = 6 m} \end{array}$
$\begin{array}{r l} & {\vec {v} _ {P Q} = \vec {v} _ {P} - \vec {v} _ {Q}} \\ & {\qquad = \left(\frac {1 0}{\sqrt {2}} \hat {\imath} + \frac {1 0}{\sqrt {2}} \hat {j}\right) - \left(- \frac {7 0}{\sqrt {2}} \hat {\imath} + \frac {7 0}{\sqrt {2}} \hat {j}\right)} \\ & {\qquad = 4 0 \sqrt {2} \hat {\imath} - 3 0 \sqrt {2} \hat {j}} \\ & {A B = 9 0 \times \frac {4}{3} = 1 2 0 m} \\ & {\Rightarrow \quad Q B = 1 0 m} \\ & {\Rightarrow \quad Q M = 1 0 \sin 3 7 ^ {\circ} = 1 0 \times \frac {3}{5} = 6 m} \end{array}$
Q60
Allen
2. NAT
Numerical
Velocity of the boat with respect to river is 10 m/s. From point A it is steered in the direction shown. Find the distance (in m) it will reach on the opposite bank from point B.
Correct Answer: 30
Explanation:
Ans. (30)
$\begin{array}{l} {t = \frac {1 2 0}{8} = 1 5 \sec} \\ {\Delta = (6 - 4) \times 1 5 = 3 0 m} \end{array}$
$\begin{array}{l} {t = \frac {1 2 0}{8} = 1 5 \sec} \\ {\Delta = (6 - 4) \times 1 5 = 3 0 m} \end{array}$
Q61
Allen
4. JEE-Advanced Pattern
MCQ
A particle is ejected from the tube at $A$ with a velocity $v$ at an angle $\theta$ with the vertical $y$ -axis. A strong horizontal wind gives the particle a constant horizontal acceleration $a$ in the $x$ -direction. If the particle strikes the ground at a point directly under its released position and the downward $y$ -acceleration is taken as $g$ then
A.
$h = \frac{2v^{2}\sin\theta\cos\theta}{a}$
B.
$h = \frac{2v^{2}\sin\theta\cos\theta}{g}$
C.
$h = \frac{2v^2}{g}\sin \theta (\cos \theta +\frac{a}{g}\sin \theta)$
D.
$h = \frac{2v^2}{a}\sin \theta (\cos \theta +\frac{g}{a}\sin \theta)$
Q62
Allen
4. JEE-Advanced Pattern
MCQ
A stone is projected from point P on the inclined plane with velocity $v_{0} = 10 \, m/s$ directed perpendicular to the plane. The time taken by the stone to strike the horizontal ground S is (Given $PO = \ell = 10 \, meter$ )
A.
1.5 sec
B.
1.4 sec
C.
2 sec
D.
2.3 sec
Q63
Allen
4. JEE-Advanced Pattern
MCQ
On a particular day rain drops are falling vertically at a speed of 5 m/s. A man holding a plastic board is running to escape from rain as shown. The lower end of board is at a height half that of man and the board makes $45^{\circ}$ with horizontal. The maximum speed of man so that his feet does not get wet, is
A.
5 m/s
B.
$5\sqrt{2}$ m/s
C.
$5/\sqrt{2}$ m/s
D.
zero
Q64
Allen
4. JEE-Advanced Pattern
MCQ
A 2 m wide truck is moving with a uniform speed of 8 m/s along a straight horizontal road. A pedestrian starts crossing the road at an instant when the truck is 4 m away from him. The minimum constant velocity with which he should run to avoid an accident is :-
A.
$1.6\sqrt{5}$ m/s
B.
$1.2\sqrt{5}$ m/s
C.
$1.2\sqrt{7}$ m/s
D.
$1.6\sqrt{7}$ m/s
Q65
Allen
4. JEE-Advanced Pattern
MSQ
The position of a particle with time is given by
$(x,y) = (8t^{2},3)$ for $t\leq t_1 = (8tt_1,3)$ for $t > t_{1}$
Choose the CORRECT alternative.
A.
Particle moves along a straight line parallel to x axis.
B.
C.
D.
Q66
Allen
4. JEE-Advanced Pattern
MSQ
The position-time $(x-t)$ graphs for two children A and B returning from their school O to their homes P and Q respectively along straight line path (taken as x axis) are shown in figure below. Choose the CORRECT statement (s):
A.
A lives closer to the school than B
B.
A starts from the school earlier than B
C.
A and B have equal average velocities from 0 to $t_{0}$ .
D.
B overtakes A on the way
Q67
Allen
4. JEE-Advanced Pattern
MSQ
A ball is thrown from ground such that it just crosses two poles of equal height kept 80 m apart. The maximum height attained by the ball is 80 m. When the ball passes the first pole, its velocity makes $45^{\circ}$ with horizontal. The correct alternatives is/are :- $(g = 10 \, m/s^{2})$
A.
Time interval between the two poles is 4 s.
B.
Height of the pole is 60 m.
C.
Range of the ball is 160 m.
D.
Angle of projection is $\tan^{-1}
Q68
Allen
4. JEE-Advanced Pattern
MSQ
A projectile is thrown with speed u into air from a point on the horizontal ground at an angle $\theta$ with horizontal. If the air exerts a constant horizontal resistive force on the projectile then select correct alternative(s).
A.
At the farthest point, the velocity is horizontal.
B.
The time for ascent equals the time for descent.
C.
The path of the projectile may be parabolic.
D.
The path of the projectile may be a straight line.
Q69
Allen
4. JEE-Advanced Pattern
MSQ
A block is thrown horizontally with a velocity of 2 m/s (relative to ground) on a belt, which is moving with velocity 4 m/s in opposite direction of the initial velocity of block. If the block stops slipping on the belt after 4 s it was dropped then choose the correct statement(s) :-
A.
Displacement with respect to ground is zero after 2.66 s and magnitude of displacement with respect to ground is 12 m after 4 s.
B.
Magnitude of displacement with respect to ground in 4 s is 4 m.
C.
Magnitude of displacement with respect to belt in 4 s is 12 m.
D.
Displacement with respect to ground is zero in 8/3 s.
Q70
Allen
4. JEE-Advanced Pattern
MSQ
A cubical box dimension L = 5/4 meter starts moving with an acceleration $\vec{a} = 0.5 m/s^{2} \hat{i}$ from the state of rest. At the same time, a stone is thrown from the origin with velocity $\vec{V} = v_{1}\hat{i} + v_{2}\hat{j} - v_{3}\hat{k}$ with respect to earth. Acceleration due to gravity $\vec{g} = 10m/s^{2}(-\hat{j})$ . The stone just touches the roof of box and finally falls at the diagonally opposite point then :
A.
$v_{1} = \frac{3}{2}$
B.
$v_{2} = 5$
C.
$v_{3}=\frac{5}{4}$
D.
$v_{3} = \frac{5}{2}$
Q71
Allen
4. JEE-Advanced Pattern
MCQ
PARAGRAPH FOR QUESTION NO. 11 TO 13
In the figure shown there is a long horizontal bridge over a river 75 m high from water surface. A strong man throws a stone in the parallel plane of the bridge. A observer in a car travelling on the bridge finds the stone going pass by the car while ascending and also while descending between two points on the road 30 m away. The car is travelling at a speed of 15 m/s. The stone is thrown from the bank of river just at the same level of water.
What is the angle the velocity makes with the bridge when it goes past the car while ascending?
What is the angle the velocity makes with the bridge when it goes past the car while ascending?
A.
$30^{\circ}$
B.
$45^{\circ}$
C.
$\tan^{-1}\left(\frac{2}{3}\right)$
D.
$\tan^{-1}\left(\frac{3}{2}\right)$
Q72
Allen
4. JEE-Advanced Pattern
MCQ
PARAGRAPH FOR QUESTION NO. 11 TO 13
In the figure shown there is a long horizontal bridge over a river 75 m high from water surface. A strong man throws a stone in the parallel plane of the bridge. A observer in a car travelling on the bridge finds the stone going pass by the car while ascending and also while descending between two points on the road 30 m away. The car is travelling at a speed of 15 m/s. The stone is thrown from the bank of river just at the same level of water.
Horizontal distance AB travelled by stone is :-
Horizontal distance AB travelled by stone is :-
A.
0 m
B.
75 m
C.
120 m
D.
240 m
Q73
Allen
4. JEE-Advanced Pattern
MCQ
PARAGRAPH FOR QUESTION NO. 11 TO 13
In the figure shown there is a long horizontal bridge over a river 75 m high from water surface. A strong man throws a stone in the parallel plane of the bridge. A observer in a car travelling on the bridge finds the stone going pass by the car while ascending and also while descending between two points on the road 30 m away. The car is travelling at a speed of 15 m/s. The stone is thrown from the bank of river just at the same level of water.
What is the distance between car and stone at the instant when particle reaches at point B?
What is the distance between car and stone at the instant when particle reaches at point B?
A.
0 m
B.
75 m
C.
120 m
D.
240 m
Q74
Allen
4. JEE-Advanced Pattern
MCQ
PARAGRAPH FOR QUESTION 14 AND 15
From the ground level, a ball is to be shot with a certain speed. Graph shows the range (R) of the particle versus the angle of projection from horizontal ( $\theta$ ).
Values of $\theta_{1}$ and $\theta_{2}$ are
Values of $\theta_{1}$ and $\theta_{2}$ are
A.
$53^{\circ}$ and $37^{\circ}$
B.
$26.5^{\circ}$ and $63.5^{\circ}$
C.
$18.5^{\circ}$ and $71.5^{\circ}$
D.
$15^{\circ}$ and $75^{\circ}$
Q75
Allen
4. JEE-Advanced Pattern
MCQ
PARAGRAPH FOR QUESTION 14 AND 15
From the ground level, a ball is to be shot with a certain speed. Graph shows the range (R) of the particle versus the angle of projection from horizontal ( $\theta$ ).
The corresponding time of flight vs $\theta$ graph is :-
The corresponding time of flight vs $\theta$ graph is :-
A.
B.
C.
D.
