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

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A stone is projected horizontally from a point $P$ , so that it hits the inclined plane perpendicularly. The inclination of the plane with the horizontal is $θ$ and the point $P$ is at a height $h$ above the foot of the incline, as shown in the figure. Determine the velocity of the projection.

Important Questions on Kinematics

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IMPORTANT

Alpha Question Bank for Engineering: Physics>Chapter 4 - Kinematics>Exercise - 4>Q 5

Two parallel straight lines are inclined to the horizontal at an angle

α .

A particle is projected from a point midway between them to graze one of the lines and strikes the other at right angles. Show that if

θ

is the angle between the direction of projection and either of lines, then

tanθ = (\sqrt{2} - 1) cotα

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Alpha Question Bank for Engineering: Physics>Chapter 4 - Kinematics>Exercise - 4>Q 6

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?

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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

A stone is projected horizontally from a point P, so that it hits the inclined plane perpendicularly. The inclination of the plane with the horizontal is θ and the point P is at a height h above the foot of the incline, as shown in the figure. Determine the velocity of the projection.

Important Questions on Kinematics

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A stone is projected horizontally from a point $P$ , so that it hits the inclined plane perpendicularly. The inclination of the plane with the horizontal is $θ$ and the point $P$ is at a height $h$ above the foot of the incline, as shown in the figure. Determine the velocity of the projection.