Range and Complementary angle of Projectile motion
Learn how projectile range varies with angle and understand the concept of complementary angles in projectile motion. Ideal for NEET, JEE Main, and JEE Advanced preparation with derivations, formulas and solved problems.
Range:
- Range of projectile motion is R =
= 
= 
. - At 450 range is maximum. As sin2
= 1 = sin 90 0. So, Rmax = 
- The maximum height for maximum range is H =
= 
= 
- Relation between horizontal range and maximum height of projectile motion: Range R =
and maximum height H = 
Now
= 
= 
= 4cot∴ R = 4Hcot
.∴ cot
= 
. If R is n times of maximum height then, R = nH = 4Hcot
.- Two particles A and B are projected from the same point with different projected velocity in such a manner that the initial vertical component of velocities of two particles remain the same. Find the ratio of range of these two particles.
Let us consider uA and uB are the projected velocities of these two particles with angle of projection 
and 
respectively with positive x axis.
We know that the range of a particle R = 
= = 
.
As the initial vertical component of velocities remain same then, 
= 
or, 
sin = 
sin
or, 
= 
Therefore, R ∝ 
Or, 
= 
= 
= 
= 
.
7. A particle is projected from a point. The particle passes two points P (a, b) and Q (b, a) during its motion. Calculate the range of the particle and angle of projection.
Let us consider, the particle is projected with speed u at an angle with horizontal.
The equation of trajectory of the particle is y = xtan[1 – 
]
When the particle passes point P then, b = atan[1 – 
] ——— (i)
When the particle passes point Q then, a = btan[1 – 
] ——— (ii)
Dividing equation (i) by equation (ii) we get, 
= 
Or, 
= 
∴ R = 
= 
.
Substituting the value of R in equation (i) we get, b = atan[1- 
]
Or, = tan[
]
Or, tan = 
∴ = 
.
Complementary angle of projection:
1. Range of projection of two particles remain same for complementary angles of projection.
If a particle is projected at an angle with horizontal with speed u then the range is R1 = .
If the particle is projected at the complementary angle i.e. (900 – ) then the range is R2 = 
= = R1.
2. Relation between maximum heights of projectiles at complementary angle:
The maximum height attained by the projectile for angle of projection is H1 = .
The maximum height attained by the projectile for angle of projection (900 – ) is H2 =
= 
.
Therefore, 
= 
= 
.
3. Relation between time of flight of projectiles at complementary angle:
If the time of flight for two projectile motions for the complementary angles and (900 – ) are T1 and T2 respectively, then T1 = 
and T2 = 
.
Therefore, T1T2 = × = 
= 
= 
.
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