Question 61: Motion in 2D & Vectors

Question

Two projectiles are thrown with same initial velocity making an angle of $$45^{\circ}$$ and  $$30^{\circ}$$ with the horizontal respectively. The ratio of their respective ranges will be :

Accepted Answer

Here's how to determine the ratio of the ranges for the two projectiles:
Understanding the Concepts
Projectile Motion:
Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory.
Range:
The range of a projectile is the horizontal distance it covers from its launch point to the point where it returns to the same vertical height.
Key Formula
The formula for the range (R) of a projectile is:
$$R = \frac{u^2 \sin 2	heta}{g}$$
where:
R = Range
u = Initial velocity (same for both projectiles)
ÃÂ¸ = Angle of projection
g = Acceleration due to gravity (constant)
Calculations
Projectile 1 (45ÃÂ° angle):
Let the range of the projectile launched at 45ÃÂ° be R
1
.
$$R_1 = \frac{u^2 \sin (2 	imes 45^{\circ})}{g} = \frac{u^2 \sin 90^{\circ}}{g} = \frac{u^2}{g} $$
Projectile 2 (30ÃÂ° angle):
Let the range of the projectile launched at 30ÃÂ° be R
2
.
$$R_2 = \frac{u^2 \sin (2 	imes 30^{\circ})}{g} = \frac{u^2 \sin 60^{\circ}}{g} = \frac{u^2 \sqrt{3}}{2g}$$
Ratio of Ranges (R
1
: R
2
):
Divide the range of projectile 1 by the range of projectile 2:
$$\frac{R_1}{R_2} = \frac{\frac{u^2}{g}}{\frac{u^2 \sqrt{3}}{2g}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$$
To simplify the ratio, multiply both numerator and denominator by Ã¢ÂÂ3:
$$\frac{R_1}{R_2} = \frac{2\sqrt{3} 	imes \sqrt{3}}{3 	imes \sqrt{3}} = \frac{6}{3\sqrt{3}} = \frac{2}{\sqrt{3}}$$
Therefore, the ratio of the ranges of the two projectiles is
2 : Ã¢ÂÂ3
, which corresponds to
Option C
.

Answer

$$1: \sqrt{2}$$

Answer

$$\sqrt{2}: 1$$

Answer

$$2: \sqrt{3}$$

Answer

$$\sqrt{3}: 2$$

Motion in 2D & Vectors

Two projectiles are thrown with same initial velocity making an angle of $45^{\circ} and 30^{\circ}$ with the horizontal respectively. The ratio of their respective ranges will be :

Choose the correct answer: