# Question #b606a

##### 1 Answer

Their launch velocities must be in a ratio of

#### Explanation:

As you know, you can break down the trajectory of an object launched at an angle **horizontal component** and a **vertical component**.

This means that you can do the same for its launch velocity,

#v_(0x) = v_0 * costheta -># horizontal component

and

#v_(0y) = v_0 * sintheta -># vertical component

Now, you know that the *maximum heights* of the two objects must be equal. You can focus solely on the *vertical component* of the movement, which is influenced by the *gravitational acceleration*,

At maximum height, the **vertical component** of the object's velocity will be equal to zero. This means that you can write

#overbrace(v_"top on y"^2)^(color(blue)(=0)) = v_(01y)^2 - 2 * g * h_1 -># for object 1

and

#overbrace(v_"top on y"^2)^(color(blue)(=0)) = v_(02y)^2 - 2 * g * h_2 -># for object 2

You know that

#v_(01y)^2 = 2 * g * h#

#h = v_(01y)^2/(2 * g) = [v_(01) * sin(theta_1)]^2/(2 * g)#

#h = (v_(01)^2 * [sin(30^@)]^2)/(2 * g) = [v_(01)^2 * (1/2)^2]/(2g) = 1/8 * v_(01)^2/g#

For the second object, you have

#h = v_(02y)^2/(2 * g) = [v_(02) * sin(theta_2)]^2/(2 * g)#

#h = (v_(02)^2 * [sin(60^@)]^2)/(2 * g) = [v_(01)^2 * (sqrt(3)/2)^2]/(2g) = 3/8 * v_(02)^2/g#

Here

The ratio between these two initial velocities will be

#1/color(red)(cancel(color(black)(8))) * v_(01)^2/color(red)(cancel(color(black)(g))) = 3/color(red)(cancel(color(black)(8))) * v_(02)^2/color(red)(cancel(color(black)(g)))#

#v_(01)^2/v_(02)^2 = 3 implies v_(01)/v_(02) = color(green)(sqrt(3))#

Notice that this is the ratio between the values of

#h = [v_(01)^2 * sin^2(theta_1)]/(2 * g) implies v_(01)^2 = (2 * g * h)/(sin^2theta_1)#

Similarly, you have

#v_(02)^2 = (2 * g * h)/(sin^2theta_2)#

Divide these expressions to get

#v_(01)^2/v_(02)^2 = color(red)(cancel(color(black)(2 * g * h)))/sin^2theta_1 * sin^2theta_2/color(red)(cancel(color(black)(2 * g * h))) = sin^2theta_2/sin^2theta_1#

This is equivalent to

#v_(01)/v_(02) = sqrt(sin^2theta_2/sin^2theta_1) = sintheta_2/(sintheta_1) = sqrt(3)/color(red)(cancel(color(black)(2))) * color(red)(cancel(color(black)(2)))/1 = color(green)(sqrt(3))#