Each year Sam looks forward to the ring toss at the county fair. Sam is not happy to hit just any peg, but instead he prides himself on hitting the middle peg in the front row each year. Suppose Sam tosses the ring with an initial horizontal velocity of 8 feet/sec and an initial vertical velocity of 3 feet/sec and that he’ll release the ring when the ring is 4 feet above the ground. Sam lines up directly in front of the middle ring at a distance of 4 feet from the ring platform. Our job is to model the motion of the ring using parametric equations and find out whether or not Sam hits his target – the middle peg. The Ring Toss game is shown below. It sits on a short stool measuring 1 foot in height. The pegs are two inches high and the bottom of the first row of pegs is 4 inches from the bottom of the game.
We will think about the horizontal and vertical positions of the ring separately and write functions for each. These functions will be written as functions of time. Let x(t) represent the horizontal position of the ring at time t and y(t) represent the vertical position of the ring at time t, where t is measured in seconds and the positions are measured in feet. These equations are called parametric equations and t is called the parameter.
Consider the horizontal position first. Since we know that Sam tosses the ring with a horizontal velocity of 8 ft/sec.
According to the passage, the peg is 2 inches high and the bottom row is 4 inches from the bottom of the game. That makes the top of the peg 6 inches from the bottom of the game. However, there is also the short stool measuring 1 foot that the game is sitting upon, so in total, the top of the peg is 1.5 feet from the ground (vertical height). Sam is also 4 feet away from the ring platform, which is the horizontal distance. This information gives me the location of the ring as (4,1.5). Sam will hit his target peg, because when I looked at my sketch of the graph, (4,1.5) is on the path of...