Question

Two balls projected at 30 degree and 60 degree with same intial velocities the ratio of their maximum heights is

Answer 1

Answer:

h₁ : h₂ ratio is 1: 3

Explanation:

Let's name :

α₁= 30° (th  angle in which the first ball was thrown)

α₂= 60° ( the angle in which the second ball was thrown)

V₁= V₂ = V (or V₀)  Their initial velocities are the same.

h₁ --> max height for ball 1

h₂ --> max height for ball 2

FORMULAS THAT WILL BE USED

1. Vy= V₀y - gt  
2. V₀y= V₀×sin α
3. h= V₀y· t - 1/2(gt²)

It's necessary to draw the situaton, so below I'm going to attach a picture of the curves of the two balls.

We must reember that Vy in the max height = 0 .  Now let's solve for h1 (max height of ball 1)

Step 1) solve for t

Vy= 0 , and V₀y= V₀×sin α , so let's replace them in the first formula.

                                                Vy= V₀y - gt

                                                0 = V₀×sin α - gt

                                                -  V₀×sin α = -gt

                                                (V₀×sin α)/g = t (1)

Step 2) replace every t in the third formula with the result in step 1

                                      h= V₀y· t - 1/2(gt²)

                                     h= V₀y· (V₀×sin α)/g - 1/2{g[(V₀×sin α)/g)²)]}

                                    h= (V₀y²sin² α)/ 2g

Step 3) Add in values for ball 1

                                    h₁=( V² sin² 30°)/ 2g

                                     h₁ = (0.25V²)/ 2g              

 since 0.25 = 1/4 , we have:           h₁= V²/ 8g

Step 4) Plug in values for ball 2

                                       h₂=( V² sin² 60°)/ 2g

                                        h₂=(0.75 V² )/ 2g          

since 0.75 = 3/4 , we have:     h₂= 3V²/ 8g

Step 5) Find the ratio ( h₁ : h₂ ratio)

                                   V²/ 8g  : 3V²/ 8g

                                    h₁ : h₂  --> 1: 3

If you want to do the h₂ : h₁  ration, then it's going to be :

                                   3V²/ 8g :   V²/ 8g

                                      h₂ : h₁ --> 3 : 1