# Numerical Problems : Kinematics

#### LEVEL – II

Q:1. Two particles are projected horizontally in opposite directions with v1 & v2 from the top of a pole. If the particles move perpendicular to each other just before striking the ground, find the height of the pole.

[Ans: v1v2 /2g ]

Q:2. A cannon fires successively two shells with velocity vo = 250 m/s; the first at the angle α1 = 60° and the second at the angle α2 = 45° to the horizontal, the azimuth being the same. Neglecting the air drag, find the time interval between firings leading to the collision of the shells.

[Ans:10.7 sec]

Q:3. An aeroplane flies in still air at a speed of 400 km/hr. Air is blowing from the south at a speed of 50 km/hr. The pilot wants to travel from point A to point B north-east of A and then to return.Calculate the direction he must steer

(a) on his onward journey (b) on his return journey.
If the distance AB is 1000 km then calculate the time taken in two journeys.

[Ans:(a) 390 56’ north of east, 2.3 hour   (b) 390 56 ‘ west of south, 2.75 hour ]

Q:4. Two particles move in a uniform gravitational field with an acceleration g. At the initial moment the particles were located at one point in space and moved with velocities v1 = 3.0 m/s and v2 = 4.0 m/s horizontally in opposite directions. Find the distance between the particles at the moment when their velocity vectors become mutually perpendicular.

[Ans:2.5 m ]

Q:5. A point moves rectilinearly with deceleration whose modulus depends on the velocity v of the particle as w = a √v , where ‘ a ‘ is a positive constant. At the initial moment the velocity of the point is equal to vo . What distance will it traverse before it stops ? What time will it take to cover that distance?

Ans: $\displaystyle (\frac{2}{3a})(v_0)^{3/2} , (\frac{2}{a}) \sqrt{v_0}$

Q:6. Find the ratio between the normal and tangential acceleration of a point on the rim of a rotating wheel when at the moment when the vector of the total acceleration of this point forms an angle of 30° with the vector of the linear velocity.

[Ans: ar/at = 1/√3]

Q:7. A fan rotates with a velocity corresponding to a frequency of 900 rev/min. When its motor is switched off, the fan uniformly slows down and performs 75 revolutions before it comes to a stop. How much time elapsed from the moment the fan was switched off to the moment it stopped ?

[Ans: 10 sec ]

Q:8. A motor cyclist, going due east with a velocity of 10 m/s, finds that the wind is blowing directly from the north. When he doubles his speed, he finds that the wind is blowing from north east. In what direction and with what velocity is the wind blowin?.

[Ans: 10√2 m/s, from north west ]

Q:9. The acceleration vector of a particle having initial speed Vo changes with distance as a = √x . Find the distance covered by the particle when its speed becomes twice the initial speed.

Ans : $\displaystyle (\frac{3V_0}{2})^{4/3}$

Q:10. An observer in a train moving with a uniform velocity finds that a car moving parallel to the train has a speed of 10 km/h in the direction of motion of the train. An object falls from the car and the observer in the train notices that the car has moved on for one minute, turned back, and moved with a speed of 10 km/h and picked up the object two minutes after turning. Find

(a) the velocity of the train relative to the ground and

(b) the velocity of the car during its forward and reverse journeys.

Assume that the object comes to rest immediately on fall from the point of view of the observer on the ground.

[Ans:(a) 3.33 km/hr (b) 13 km/hr, 6.67 km/hr ]

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