Question

A ball is projected from a point with speed v and angle of elevation 0₁. Another ball is projected with the same speed v and from the same point but with angle of elevation 82, where tan0₂ =0.5 tan0₁. If R₁, h₁, and R₂, h2 are respectively the ranges and maximum heights and R₁ R₂ = 8:5. Find the ratio h1:h₂.​

Answer 1

h₁ / h₂ = (8/5) * ((1 + tan^2(θ₂)) / (1 + tan^2(θ₁)))

Step-by-step explanation:

Answer 2

Step-by-step explanation:

Let's denote the initial speed of both balls as v and the angle of elevation of the first ball as θ₁ and the angle of elevation of the second ball as θ₂.

Given that tan(θ₂) = 0.5 * tan(θ₁), we can express θ₂ in terms of θ₁ as follows:

tan(θ₂) = 0.5 * tan(θ₁)

θ₂ = arctan(0.5 * tan(θ₁))

Now, let's analyze the motion of the first ball (θ₁) and the second ball (θ₂):

For the first ball (θ₁):

Range (R₁) = (v² * sin(2θ₁)) / g

Maximum height (h₁) = (v² * sin²(θ₁)) / (2 * g)

For the second ball (θ₂):

Range (R₂) = (v² * sin(2θ₂)) / g

Maximum height (h₂) = (v² * sin²(θ₂)) / (2 * g)

Now, we know that R₁ / R₂ = 8 / 5. Therefore:

(v² * sin(2θ₁)) / (v² * sin(2θ₂)) = 8 / 5

Since v is common to both equations, we can cancel it out:

sin(2θ₁) / sin(2θ₂) = 8 / 5

Using the double angle formula for sine, sin(2θ) = 2 * sin(θ) * cos(θ), we can rewrite the above equation as:

2 * sin(θ₁) * cos(θ₁) / 2 * sin(θ₂) * cos(θ₂) = 8 / 5

Now, we know that tan(θ₂) = 0.5 * tan(θ₁), and using the identity tan(θ) = sin(θ) / cos(θ), we can express cos(θ₂) in terms of cos(θ₁):

cos(θ₂) = cos(θ₁) / (2 * sin(θ₁))

Substitute this into the equation:

2 * sin(θ₁) * cos(θ₁) / [2 * sin(θ₂) * (cos(θ₁) / (2 * sin(θ₁)))] = 8 / 5

Now, simplify the expression:

sin(θ₁) * cos(θ₁) / [sin(θ₂) * cos(θ₁) / sin(θ₁)] = 8 / 5

sin²(θ₁) / sin(θ₂) = 8 / 5

Now, using the maximum height equation for the first ball (h₁), we can express sin²(θ₁) in terms of h₁:

sin²(θ₁) = (2 * g * h₁) / v²

Substitute this back into the equation:

(2 * g * h₁) / v² / sin(θ₂) = 8 / 5

Now, rearrange the equation to solve for h₁ / sin(θ₂):

h₁ / sin(θ₂) = (8 / 5) * (v² / 2 * g)

Now, let's analyze the height equation for the second ball (h₂) using θ₂:

h₂ = (v² * sin²(θ₂)) / (2 * g)

Substitute sin(θ₂) = 0.5 * tan(θ₁) into the equation:

h₂ = (v² * (0.5 * tan(θ₁))²) / (2 * g)

h₂ = (v² * 0.25 * tan²(θ₁)) / (2 * g)

h₂ = (v² * tan²(θ₁)) / (8 * g)

Now, divide the equation for h₁ / sin(θ₂) by the equation for h₂:

(h₁ / sin(θ₂)) / h₂ = [(8 / 5) * (v² / 2 * g)] / [(v² * tan²(θ₁)) / (8 * g)]

Cancel out v² and g:

(h₁ / sin(θ₂)) / h₂ = (8 / 5) / (2 * tan²(θ₁) / 8)

(h₁ / sin(θ₂)) / h₂ = (8 / 5) / (tan²(θ₁) / 4)

Now, simplify the expression:

(h₁ / sin(θ₂)) / h₂ = (8 / 5) * (4 / tan²(θ₁))

(h₁ / sin(θ₂)) / h₂ = 32 / (5 * tan²(θ₁))

Now, recall that tan(θ₂) = 0.5 * tan(θ₁), so tan²(θ₁) = 4 * tan²(θ₂).

Substitute this back into the equation:

(h₁ / sin(θ₂)) / h₂ = 32 / (5 * 4 * tan²(θ₂))

(h₁ / sin(θ₂)) / h₂ = 32 / (20 * tan²(θ₂))

(h₁ / sin(θ₂)) / h₂ = 8 / (5 * tan²(θ₂))

Finally, we know that R₁ / R₂ = 8 / 5, which also implies h₁ / h₂ = 8 / 5.

So, the ratio h₁ : h₂ is 8 : 5.