Relative velocity in two dimensions

Relative velocity:

Velocity of one body with respect to that of another body is called Relative velocity

1. The velocity of `A` with respect to that of `B` is given by 

2. If ‘A’ and ‘B’ are moving at an angle with one another, then,   velocity of `A` with respect to that of `B` is given by   

Projectile:

A body projected in to air with some velocity at an angle (other than 90o) with the horizontal is called Projectile.

Examples:

(i) A body dropped from the window of a moving train

(ii) A bullet fired from the gun

(iii) An athlete doing the high jump

(iv) A cricket ball thrown in any direction

(v) A hammer thrown by an athlete.

1. Path of Projectile:

If a body is thrown with an initial velocity ‘U’ making an angle   with the horizontal

‘U’ is resolved as

Horizontal displacement covered by the body in time ‘t’

Vertical displacement covered by the body in time ‘t’,

This is the equation of parabola. Thus the trajectory of a projectile is a parabola.

2. Maximum height & horizontal range:

Maximum Height (h_max):

At maximum height, V = 0

Horizontal Range (R):

Horizontal range = Horizontal component of velocity X Time of flight.

When a projectile is thrown at an angle 45⁰ with the horizontal the range becomes maximum.

For the projectile, two angles of projection for the same range are θ and (90-θ ) & (45+θ ) and (45-θ )

If the range and maximum height of a projectile are equal, the angle of projection is 

3. Relation between maximun height & horizontal range:

If a projectile thrown with velocity ‘u’ making an angle ‘θ’  with the horizontal.

Time of ascent , Time of descent & Time of flight: