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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The benches of a gallery in a cricket stadium are $1 m$ wide and $1 m$ high. A batsman strikes the ball at a level of $1 m$ above the ground and hits a mammoth sixer. The ball starts at $35 m s^{- 1}$ at an angle of $53 °$ with the horizontal. The benches are perpendicular to the plane of motion and the first bench is $110 m$ from the batsman. On which bench will the ball hit?

Important Questions on Kinematics

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Alpha Question Bank for Engineering: Physics>Chapter 4 - Kinematics>Exercise - 4>Q 7

A ship is approaching a cliff of height

105 m

above sea level. A gun fitted on the ship can fire shots with a speed of

110 m s^{- 1}

. Find the maximum distance from the foot of the cliff from where the gun can hit an object on the top of the cliff.

[g = 10 m s^{- 2}]

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Alpha Question Bank for Engineering: Physics>Chapter 4 - Kinematics>Exercise - 4>Q 8

Shots fired simultaneously from the top and bottom of a vertical cliff with the elevation

α

and

β

respectively, strike an object simultaneously at the same point on the ground. If

s

is the horizontal distance of the object from the cliff, then what is the height of the cliff.

(β > α)

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Alpha Question Bank for Engineering: Physics>Chapter 4 - Kinematics>Exercise - 4>Q 9

Some students are playing cricket on the roof of a building of height

20 m

. While playing, a ball falls on the ground. A person on the ground returns their ball with the minimum possible speed at angle of projection

45 °

with the horizontal. The speed of projection is

20 \sqrt{α} m s^{- 1}

. Here

α

is an integer. Find

α

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Alpha Question Bank for Engineering: Physics>Chapter 4 - Kinematics>Exercise - 4>Q 10

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})

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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.)

Question Image

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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?

Question Image

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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).

Important Questions on Kinematics

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