Question 26 This question is a long free-response question. Show your work for each part of the question.

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 v0 in the +x 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 x and other quantities. After examining the equation, the student claims that y is proportional to x2 .

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

Answer 1

1. Calculate the time required for the sphere to fall to the ground when dropped with no vertical downward velocity is determined by solving the following equation for t (time): y = 0.5 x g x t². This rearranges to

t = √( y/0.5 x g) where g is the acceleration of gravity.

2 . Substitute the t in the equation in step 1 into x = v_o • t yielding x = v_o • √( y/0.5 x g).

3. Square both sides of the equation x² = v_o² • (y/(0.5 • g). Rearranging y = x²• g /(2• v_o²⁾.