Motion In A Plane

In absence of any external force, linear momentum of a system is conserved..
Linear momentum of bullet-gun system is conserved.
i.e.. Initial momentum = Final momentum

or

Negative sign implies recoil.
Hence, the velocity of recoil is .

Centripetal force is given by

where m is mass of particle, is speed, and R is radius of circular path.

or
Given,

or
Therefore, centripetal force will become half.

As seen from the cart the projectile moves vertically upwards and comes back.
The time taken by cart to cover

Here,
(for a projectile going upwards)
and
From first equation of motion

Two vectors (inclined at any angle) and their sum vector form a triangle.
It is given that two vectors have a resultant equal to either of them, hence these three vectors form an equilateral triangle each angle of .
In the figure and are two vector having their sum vectors such that.

Thus, the vectors and of same magnitude have the resultant vectors of the same magnitude.
In this case angle between and is
Alternative : Let there be two vectors and
where,
Their sum is

Taking self product of both sides, we get



where is angle between and
When , then we have




In this condition angle between given vectors should be .

Let a body be projected at a velocity at an angle with the horizontal. Then horizontal range covered is given by

(i)
and height is
(ii)
Given

Also,




Hence, angle of projection is

Key Idea Horizontal ranges are same for complementary angles of projection ie, for and
We know that if two stones have same horizontal range, then this implies that both are projected at and
Here,

For first stone, Max. height
For second stone, Max. height,

or