Question

Four particles are fired with the same velocities
at angles 20°,40°,55º and 70° with the
horizontal. The range of pojectile will be the
largest for the one projected at angle:​

Answer 1

Answer : The range of pojectile will be the largest for the one projected at angle 40 degrees

The question contains angles such as 55,70, 40, 20 for which Sine isn't a definite value or known value. So, This question is entirely based on Conceptual knowledge.

First of all, sine is an increasing function in first quadrant.

$\bold{ \sin(0 ^{ \circ} ) = 0} \\ \bold{\sin(90 ^{ \circ} ) = 1}$

So, Sine function increases from 0 to 1, as the angle increases.

From the above knowledge one can say

$\sin70 ^{ \circ} > \sin55 ^{ \circ} > \sin40 ^{ \circ} > \sin20 ^{ \circ}$

Given that, Four particles were fired with same velocity. At a place, Acceleration due to gravity is constant for all four particles.

Now,

$R = \frac{u ^{2} \: \sin(2 \theta) }{g}$

Since u, g are constant for all four particles, We can say

$R \propto \: \sin 2\theta$

We can say, The angle will the greatest value of sin2θ will have the greatest range.

Now time for a bit of mathematics.

sin(2*70) = sin(140) = sin40

sin(2*55)= sin(110) = sin70

sin(2*40) = sin80

sin(2*20) = sin40

[ Since, sin2θ is same for both 70, 20 degrees, They will have the same range. Thus, There is a same range for angles which are complementary]

From our earlier discussion,

sin80 > sin70 > sin40 = sin40

Thus, The range is maximum for the particle projected with an angle 40 degrees.

Answer 2

The range of pojectile will be the

largest for the one projected at angle 40 ^°