Warm UpMay 8 th 1) Determine whether or not (104, -200) is a point on the graph of x = 4 + t, y = t. 2) Imagine you are piloting a small plane at an altitude of 15,000 feet and preparing to land. Once you begin your descent to the runway, your altitude changes at a rate of-25 feet/sec. Your horizontal speed is 180 ft /sec. Write parametric equations to model the descent of your plane. 3)The function y = f (x) is defined parametrically by x(t) = 1 + 3t, y(t) = 2t Write y as a function of x.

Homework Questions?

Parametric Equations & Projectile Motion

Parametric Equations for Projectile Motion If an object starts at (x 0, y 0 ) at t = 0 with initial velocity v 0 in the direction θ, then its position at time t is given by

Kevin hits a baseball from 3 feet above the ground with an initial speed of 150 ft/sec at an angle 18º with the horizontal. Write parametric equations to simulate the movement of the ball. Will the ball clear a 20 ft. wall that is 400 feet away?

The regulation height of a basketball hoop is 10 feet. Let the location of the basket be represented in the coordinate plane by the point (0, 10). Let the ball be thrown at a 45° angle with the ground. Suppose Nancy is standing 10 feet from the basket by the point (-10, 0), and she shoots a basket from 6 feet in the air with an initial velocity of 22 ft /s. Write parametric equations that represent the ball’s motion through the air. Will Nancy make the basket? Defend your reasoning.

Experiment on your calculator with different direction angles until Nancy makes a basket. What angle did you use?

Suppose Nancy is shooting from the free throw line located 15 feet from the basket and always releases the ball at a 45º angle from a height of 6 feet. Let the ball be thrown with an initial velocity of 22 feet per second. Write parametric equations to represent the ball’s motion through the air. Will she make the basket? Defend your reasoning.

Experiment on your calculator with different initial velocities until Nancy makes a basket. What velocity did you use?

Nigel and Nancy go to an amusement park. They find a carnival game that combines Nancy’s love of basketball and Nigel’s interest in Ferris wheels. Baskets are mounted on a rotating wheel. The players have to toss a basketball into the moving basket. If they get the ball in the blue basket, they win a big prize. The edge of the wheel faces the player and rotates towards the player (counterclockwise). The diagram shows a side view of the game. Let t = 0 be the time when the blue basket is at the top of the Ferris wheel. The center of the wheel is (0,6). The wheel completes one revolution every 15 seconds. Write parametric equations to represent the movement of the wheel.

What are the coordinates of the basket when it has completed one-fourth of a revolution? How long does it take the basket to get to that point?

The player stands 15 feet away from the center of the wheel at (-15, 0). Nigel decides to aim for the basket when it is at the point (-4, 6). He releases the ball from 6 feet at a 45° angle. Explain why parametric equations to represent the location of the basketball as a function of time since the ball was thrown are:

If Nigel throws with an initial velocity of 22 ft /sec, will the path of the basketball pass through the point (-4, 6)? Explain. Determine an initial velocity that will guarantee the path of the basketball will pass through the point (-4, 6). How long does it take the basketball to get to the point (-4, 6)?

How long should Nigel wait after he sees the blue basket at the top of the wheel before shooting the basket? Adjust your basketball parametric equations so the ball and the blue basket meet at the point (-4, 6) at the same time. Check your results on your graphing calculator, graphing both the basketball and wheel parametric equations. Write the equations you used.