Question

. A particle is projected with a certain velocity so as to pass through a point at a horizontal distance R
from the point of projection. Show that if t1 and t2 are the time taken to reach this point in two
possible ways in which the particle can be projected with the given velocity then t1t2 = R/g​

Answer 1

Answer:

The particle will follow two equations:

r=vcosθt

and t=

g

2vsinθ

⟹

v

2

t

2

r

2

+

4v

2

t

2

g

2

=1

⟹g

2

t

4

−4v

2

t

2

+4r

2

=0

This is a quadratic equation in t

2

with its solutions t

1

2

,t

2

2

. For a quadratic equation, ax

2

+bx+c=0, product of roots is given by c/a

⟹t

1

2

t

2

2

=

g

2

4r

2

⟹t

1

t

2

=

g

2r

∝r