A body is thrown at an angle θ 0 with the horizontal such that it attains a speed equal to 2 3 times the speed of projection when the body is at half of its maximum height. Find the angle θ 0 .

Let velocity of projection = v 0 Given that angle of projection θ 0 At any height y , Velocity in y-direction by using first equation of motion v y = v 0 sin θ 0 2 - 2 g y And Velocity in x-direction v x = v 0 cos θ 0 The speed of the projectile is v = v x 2 + v y 2 ⇒ v = v 0 sin θ 0 2 - 2 g y + v 0 cos θ 0 2 ⇒ v = v 0 2 - 2 g y . . . . 1 where y = y max 2 and y max = v 0 2 sin 2 θ 0 2 g ,Substituting in equation 1 ⇒ v = v 0 2 - 2 g v 0 2 sin 2 θ 0 4 g We have v = v 0 1 - sin 2 θ 0 2 Since v = 2 3 v 0 (given), we have 1 - sin 2 θ 0 2 = 2 3 This gives sin θ 0 = 2 3 . Hence, θ 0 = sin - 1 2 3 .

EASY

JEE Advanced

IMPORTANT

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A particle is projected with velocity $u$ at angle $θ$ with horizontal. Calculate the time when it is moving perpendicular to initial direction. Also calculate the velocity at this position.

Important Questions on Kinematics II

HARD

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 5 - Kinematics II>ILLUSTRATION 5.33>Q (a)

Two inclined planes $O A$ and $O B$ having inclination (with horizontal) $30 °$ and $60 °$ , respectively, intersect each other at $O$ as shown in the figure. A particle is projected from point $P$ with velocity $u = 10 \sqrt{3} m s^{- 1}$ along a direction perpendicular to plane $O A$ . If the particle strikes plane $O B$ perpendicularly at $Q$ , calculate the

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(a) velocity with which the particle strikes the plane $O B$ .
(b) time of flight.
(c) vertical height $h$ of $P$ from $O$ .
(d) maximum height from $O$ , attained by the particle.
(e) distance $P Q$ .

MEDIUM

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 5 - Kinematics II>ILLUSTRATION 5.34

A truck starts from origin accelerating with

a {m s}^{- 2}

in positive

x

-axis direction. After

2 s

, a man standing at the starting point of the truck projected a ball at an angle

30 °

with velocity

v

m s^{- 1}

. Find the relation between

a

and

v

such that the ball hits the truck. (Assume that truck is moving on horizontal plane and the man projected the ball from the same horizontal level of truck).

HARD

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 5 - Kinematics II>ILLUSTRATION 5.35

A body is thrown at an angle

θ_{0}

with the horizontal such that it attains a speed equal to

\sqrt{\frac{2}{3}}

times the speed of projection when the body is at half of its maximum height. Find the angle

θ_{0}

MEDIUM

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 5 - Kinematics II>ILLUSTRATION 5.36

A mango in a tree is located at a horizontal and vertical distance of

30 m

and

40 m

, respectively, from the point of projection of a stone. Find the minimum speed and the critical angle of projection of the stone so as to hit the mango.

MEDIUM

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 5 - Kinematics II>CONCEPT APPLICATION EXERCISE 5.3>Q 1

A ball rolls off top of a staircase with a horizontal velocity

u m s^{- 1}

. If the steps are

h

m

high and

b

m

wide, the ball will just hit the edge of

n^{th}

step. Find the value of

n

EASY

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 5 - Kinematics II>CONCEPT APPLICATION EXERCISE 5.3>Q 2

A body is projected horizontally from the top of a tower with initial velocity

18 m s^{- 1}

. It hits the ground at angle

45 °

. What is the vertical component of velocity when it strikes the ground?

MEDIUM

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 5 - Kinematics II>CONCEPT APPLICATION EXERCISE 5.3>Q 3

A man standing on the roof of a house of height

h

throws one particle vertically downwards and another particle horizontally with the same velocity

u

. Find the ratio of their velocities when they reach the earth's surface.

MEDIUM

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 5 - Kinematics II>CONCEPT APPLICATION EXERCISE 5.3>Q 4

A motorcycle stunt rider rides off the edge of a cliff. Just at the edge, his velocity is horizontal, with magnitude

9.0 m s^{- 1}

. Find the motorcycle's position, distance from the edge of the cliff and velocity after

0.5 s

A particle is projected with velocity u at angle θ with horizontal. Calculate the time when it is moving perpendicular to initial direction. Also calculate the velocity at this position.

Important Questions on Kinematics II

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A particle is projected with velocity $u$ at angle $θ$ with horizontal. Calculate the time when it is moving perpendicular to initial direction. Also calculate the velocity at this position.