A small cannon A is mounted on a platform (which can be rotated so that the cannon can aim at any point) and is adjusted for the maximum range R of shells. It can throw shells at any point on the shown circle (dotted) on the ground. Suddenly a wind storm starts blowing in a horizontal direction normal to A B with a speed 2 times the velocity of the shell. At what least distance can the shell land from point B . (Assume that the velocity of the wind storm is imparted to the shell in addition to its velocity of projection. Also, assume that the platform is kept stationary while projecting the shell.)

Let the speed of projection of shell from the cannon is u and that of wind is v . According to question, v = 2 u . Time of flight of the shell is given as T = 2 u sin θ g . For maximum range of projectile, angle of projection is θ = 45 ° . ⇒ T = 2 u sin 45 ° g = 2 u g . Horizontal component of velocity of shell in x and y axis is given as V x = 2 u and V y = u cos 45 ° = u 2 . Total distance covered by shell in x and y axis is given as S x = 2 u 2 u g = 2 u 2 g and S y = u 2 2 u g = u 2 g . The maximum range of the projectile is given as R = u 2 g . S x = 2 R and S y = R . Hence, shell will land at a distance 2 R from B in the direction of wind.

A body A falls freely from some altitude H ( < < R e ) . At the moment the first body starts falling another body B is thrown from the earth's surface, which collides with the first at an altitude h = H 2 . The horizontal distance of that point of collision is l from the starting point of B . Find the initial velocity and the angle at which it was thrown?

Here, Let us consider both bodies collide at time t after their projection. For body A :- Body A drops and collides with body B at point C at time t . Then, vertical displacement of body A is given as y = u t + 1 2 a t 2 ⇒ y A = - H - h = - H 2 = - 1 2 g t 2 ⇒ t = H g . . . ( i ) For body B :- Body B is projected from ground and meets body A at point C after time t . Then, Horizontal Displacement : x B = l = v x t = v cos α t . . . ( ii ) Vertical Displacement: y B = h = H 2 = v y t - 1 2 g t 2 = v sin α t - 1 2 g t 2 . . . ( iii ) Using equation ( i ) in ( iii ) , we get ⇒ H = v sin α t . . . ( iv ) Dividing equation ( iv ) by ( ii ) , we get ⇒ tan α = H l . . . ( v ) ⇒ α = tan - 1 H l Using equation ( i ) and ( v ) in ( iv ) , we get H = v H H 2 + l 2 H g ⇒ H 2 = v 2 H 2 H 2 + l 2 H g ⇒ v = g H 1 + l 2 H 2 m s - 1

A small ball rolls of the top of a stairway horizontally with a velocity of 4 . 5 m s - 1 . Each step is 0 . 2 m high and 0 . 3 m wide. If g is 10 m s - 2 , then the ball will strike the n t h step where n is equal to (assume ball strike at the edge of the step).

As given, initial velocity : u x = 4 . 5 m s - 1 and u y = 0 If the ball strikes the n t h step in time t then : Horizontal displacement : x = 0 . 3 n = u x t ⇒ 0 . 3 n = 4 . 5 t . . . ( i ) Vertical Displacement : y = 0 . 2 n = u y t - 1 2 g t 2 = 0 - 1 2 g t 2 . . . ( ii ) Dividing equation ( i ) by ( ii ) , we get 3 2 = 9 10 t ⇒ t = 3 5 sec Then, using i , we have 0 . 3 n = 4 . 5 t = 4 . 5 × 3 5 ∴ n = 9

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A cannon fires successively two shells with velocity $v_{0} = 250 m s^{- 1};$ the first at the angle $θ_{1} = 53 °$ and the second at the angle $θ_{2} = 37 °$ to the horizontal, in the same vertical plane, neglecting the air drag, find the time interval (in $\sec$ ) between firings leading to the collision of the shells. $(g = 10 m s^{- 2})$

Important Questions on Kinematics

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Alpha Question Bank for Engineering: Physics>Chapter 4 - Kinematics>Exercise - 4>Q 11

A small cannon $A$ is mounted on a platform (which can be rotated so that the cannon can aim at any point) and is adjusted for the maximum range $R$ of shells. It can throw shells at any point on the shown circle (dotted) on the ground. Suddenly a wind storm starts blowing in a horizontal direction normal to $A B$ with a speed $\sqrt{2}$ times the velocity of the shell. At what least distance can the shell land from point $B$ . (Assume that the velocity of the wind storm is imparted to the shell in addition to its velocity of projection. Also, assume that the platform is kept stationary while projecting the shell.)

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Alpha Question Bank for Engineering: Physics>Chapter 4 - Kinematics>Exercise - 4>Q 12

A body $A$ falls freely from some altitude $H (< < R_{e})$ . At the moment the first body starts falling another body $B$ is thrown from the earth's surface, which collides with the first at an altitude $h = \frac{H}{2}$ . The horizontal distance of that point of collision is $l$ from the starting point of $B$ . Find the initial velocity and the angle at which it was thrown?

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Alpha Question Bank for Engineering: Physics>Chapter 4 - Kinematics>Exercise - 4>Q 13

Two guns situated on the top of a hill of height

10 m

fire one shot each with the same speed

5 \sqrt{3} m s^{- 1}

at some interval of time. One gun fires horizontally and other fires upwards at an angle of

60 °

with the horizontal. The shots collide in the air at point

P

. Find

: -

(a) .

The time interval between the firings and

(b) .

The coordinates of the point

P

. Take the origin of the coordinate system at the foot of the hill right below the muzzle and trajectories in

x - y

plane.

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Alpha Question Bank for Engineering: Physics>Chapter 4 - Kinematics>Exercise - 4>Q 14

A small ball rolls of the top of a stairway horizontally with a velocity of

4.5 m s^{- 1}

. Each step is

0.2 m

high and

0.3 m

wide. If

g

10 m s^{- 2}

, then the ball will strike the

n^{t h}

step where

n

is equal to (assume ball strike at the edge of the step).

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Alpha Question Bank for Engineering: Physics>Chapter 4 - Kinematics>Exercise - 4>Q 15

A particle moves along the parabolic path

x = y^{2} + 2 y + 2

, where

x

and

y

are in

m

, in such a way that the

y

-component of velocity vector remains

5 m s^{- 1}

during the motion. The magnitude of the acceleration of the particle (in

m s^{- 2}

) is:

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Alpha Question Bank for Engineering: Physics>Chapter 4 - Kinematics>Exercise - 4>Q 16

A building

4.8 m

high

2 b meters

wide has a flat roof. A ball is projected from a point on the horizontal ground

14.4 m

away from the building along its width. If projected with velocity

16 m s^{- 1}

at an angle of

45 °

with the ground, the ball hits the roof in the middle, find the width

2 b

. Also, find the angle of projection so that the ball just crosses the roof if projected with velocity

10 \sqrt{3} m s^{- 1}

(g = 10 m s^{- 2})

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Alpha Question Bank for Engineering: Physics>Chapter 4 - Kinematics>Exercise - 4>Q 17

A vertical pole has a red mark at some height. A stone is projected from a fixed point on the ground. When projected at an angle of

45 °

it hits the pole orthogonally

1 m

above the mark. When projected with a different speed at an angle of

\tan^{- 1} (\frac{3}{4}),

it hits the pole orthogonally

1.5 m

below the mark. Find the speed and angle of projection so that it hits the mark orthogonally to the pole.

[g = 10 m \sec^{- 2}]

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Alpha Question Bank for Engineering: Physics>Chapter 4 - Kinematics>Exercise - 4>Q 18

A small ball is thrown from the edge of one bank of a river of width $100 m$ to just reach the other bank. The ball was thrown in the vertical plane (which is also perpendicular to the banks) at an angle $37 º$ to the horizontal. Taking the starting point as the origin $O$ , vertically upward direction as positive $y$ -axis and the horizontal line passing through the point $O$ and perpendicular to the bank as $x$ -axis find

(a) equation of trajectory of the image formed by refraction by the water surface (water surface is at the level $y = 0$ )

(b) The instantaneous velocity of the image formed due to refraction. $(g = 10 m s^{- 2}, n_{water} = \frac{4}{3})$ .

Important Questions on Kinematics

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