Question

Students are studying the two-dimensional motion of objects as they move through the air. Specifically, they are examining the behavior of a sphere that is launched horizontally from a location above the floor with an initial velocity vo in the +3 direction, as shown in the figure. The students assume that the positive directions are along the sphere's initial velocity for horizontal motion and downward for vertical motion.
The horizontal displacement of the object from its starting point is x, and the vertical displacement of the object from its starting point is y. One of the students derives an equation for y in terms of xx and other quantities. After examining the equation, the student claims that y is proportional to x^2.

Required:
Derive an equation for the vertical coordinate y of the sphere as a function of x, v0, and physical constants, as appropriate.

Answer 1

Answer:

   y = - (½ g / v₀²)   x²

Explanation:

This is a projectile launch exercise where there is no acceleration on the x-axis so

        x = v₀ₓ t

        v₀ₓ = v₀ cos tea

        y = $v_{oy}$ t - ½ g t2

        v_{oy} = v₀ sin θ

as the sphere is thrown horizontally, the angle is tea = 0º, so the initial velocity remains

          v₀ₓ = v₀

           v_{oy} = 0

we substitute in our equations

          x = v₀ t

          y = - ½ g t²

we eliminate the time from these equations, we substitute the first in the second

      y = - ½ g (x / v₀)²

      y = - (½ g / v₀²)   x²

this is the equation of a parabola