If two stones projected from the same point with the same speed but at angles pi/3 and pi/6 respectively have their ranges R₁ and R₂ then a)R1=2R2 B)R1=R2 C)R1=5R2 D)R1=25R2
To solve this problem, you can apply the formula for range for a projectile motion.
The range (R) of a projectile is given by the formula:
R = (v^2 * sin(2θ))/g
where:
- v is the initial velocity of the projectile
- θ is the angle of projection
- g is the acceleration due to gravity
In this problem, we are given that the initial speeds of both stones are the same. Therefore, v is the same for both stones.
Let's denote the angles of projection as θ₁ = π/3 and θ₂ = π/6. We need to find the relationship between R₁ and R₂.
For Stone 1:
R₁ = (v^2 * sin(2θ₁))/g
For Stone 2:
R₂ = (v^2 * sin(2θ₂))/g
Now, let's substitute the values of θ₁ and θ₂ into the equations:
R₁ = (v^2 * sin(2(π/3)))/g
R₂ = (v^2 * sin(2(π/6)))/g
Simplifying these equations using trigonometric identities:
R₁ = (v^2 * sin(π/3))/g = (v^2 * √3/2)/g
R₂ = (v^2 * sin(π/6))/g = (v^2 * 1/2)/g
Now, we can compare the relationships between R₁ and R₂:
R₁/R₂ = [(v^2 * √3/2)/g] / [(v^2 * 1/2)/g]
= (v^2 * √3/2) / (v^2 * 1/2)
= √3/1
= √3
Since R₁/R₂ = √3 and all other options given in the answer choices have ratios that are different from √3, we can conclude that the correct answer is:
D) R₁ = 25R₂
The equation for range is
R(θ)=v²sin(2θ)/g
Examine the values of sin(2θ) to arrive at the correct ratio .