Hello, my problem is the next one:

There are three objects that are launched at the Same time But with different angles( they describe proyectile motion ) . All the objects got the Same maximun height. My question is: Is the Initial velocity the Same for the three objects? Why or Why not?
Please help

Y^2 = Yo^2 + 2g*h.

Yo = sqrt(Y^2-2g*h).
So if each object has the same value of Y and h, their initial Y component of velocity would be equal.

We have 3 variables:
1. Y = 0 = Vertical component of final velocity.

2. Yo = Vertical component of initial velocity.

3.The max height(h).

To determine whether the initial velocity is the same for three objects launched at the same time but with different angles, we need to consider the factors that affect the maximum height achieved by a projectile.

In projectile motion, the maximum height attained depends on the initial vertical velocity (Vy) and the acceleration due to gravity (g). It is important to note that the initial vertical velocity can be decomposed into two components: Vy = V * sin(θ), where V represents the initial velocity magnitude and θ is the launch angle.

Now, let's consider the three objects independently:

1. Object 1: Launched with angle θ1 and initial velocity V1. Its maximum height is h1, and its initial vertical velocity is Vy1 = V1 * sin(θ1).
2. Object 2: Launched with angle θ2 and initial velocity V2. Its maximum height is h2, and its initial vertical velocity is Vy2 = V2 * sin(θ2).
3. Object 3: Launched with angle θ3 and initial velocity V3. Its maximum height is h3, and its initial vertical velocity is Vy3 = V3 * sin(θ3).

Now, to determine if the initial velocities are the same for all three objects, we need to evaluate whether Vy1 = Vy2 = Vy3.

Since we know that all three objects reach the same maximum height, h1 = h2 = h3, we can use this information to compare the initial vertical velocities.

If h1 = h2 = h3, it implies that the total time of flight for the three objects is the same, as maximum height is achieved halfway through the total time of flight.

Given that the total time of flight depends on the initial vertical velocity, it is evident that for h1 = h2 = h3, the initial vertical velocities should also be equal.

Therefore, considering that Vy1 = Vy2 = Vy3, we can conclude that the initial velocities of the three objects (V1, V2, and V3) must differ if their launch angles (θ1, θ2, and θ3) are not equal.

In conclusion, the initial velocities are not the same for the three objects because the launch angles differ, and the initial vertical velocities are dependent on both the initial velocity magnitude and the launch angle.