5.9.2 – Vertical Motion

• Recall, a parabola, or equation of a parabola, may model the path of different objects

Vertical Motion Model

• Since most objects travel against gravity according to a parabolic motion, we can use a quadratic to help us model this

• There are 2 scenarios we will consider;• 1) If a person simply drops an item from some pre-determined height• 2) If a person launches or throws an item from some pre-determined

height

Types of Models

• For case 1, where an item is dropped form a height h;• The height of the object at time t is modeled by, h(t) = -16t2 + h0

• For case 2, where an item is tossed with some type of vertical motion from a height h and a velocity v;• The height of the object at time t is modeled by, h(t) = -16t2 + v0t + h0

Time, Height, and Velocity

• For either case, we will also measure time in seconds• Accelration due to gravity is in seconds

• Our initial height should be in feet• Sometimes, we may need to convert

• Velocity will always be expressed as feet/second (this will always be given to us)

Writing an Equation

• Example. A basketball is shot in the air from the free-throw line. The ball has an initial height of 6 feet. When the ball leaves the hand of the player, it has an upward velocity of 20 feet per second.

• Write an equation for the vertical motion of the basketball.

Calculator portion

• Example. Using the equation we found for the motion, find the following information.

• 1) What is the maximum height of the ball? When does it reach the maximum height?

• 2) If the ball goes perfectly through the hoop (does not bank off the backboard), when does the ball hit the ground?

• Example. A person stands on top of a 2-story balcony. Below them, a friend asks them to drop a football down to them. If the height of the balcony is 25 feet, find the following information.

• A) Write an equation to model the motion of the ball.

• B) Find the time it takes the ball to reach the ground.

• Using the information we know about vertical motion, write your own problem for each scenario;

• 1) Dropping an object directly to the ground• 2) Throwing an object with an upward motion/initial velocity

• Using your calculators, after your write the problem, locate • 1) The maximum height of your object• 2) The time it takes to reach the ground

• Assignment• Pg. 279• 63-69 all

• Today we are going to review the topics for the test. Will post a practice test today to review tomorrow.

• Topic 1; Simplifying square roots

• 1) √32

• 2) √100

• 3) √15

• Topic 2; Solving using square roots

• 1) 2x2 = 8

• 2) 3x2 + 4 = 28

• 3) 4x2 = -16

• Topic 3; Complex Numbers

• 1) (2 + 4i) + (3 + 5i)

• 2) (2 + 4i) – (3 + 5i)

• 3) (2 + 4i)(3 + 5i)

• 4) (2 + 4i)/(3 + 5i)

• Topic 4; Complex Conjugates

• Find the conjugates for;

• 1) 3 + 9i• 2) 4 – 10i• 3) 10i + 3• 4) 9i

• Topic 5, Quadratic Formula and the Discriminant

• Given the equation y = x2 – 12x + 20, find

• 1) The number of solution(s) and type(s)

• 2) The solutions using the quadratic formula

• Topic 6; Vertical Motion Models

• 1) You throw a tennis ball in the air to serve, but misjudge it and decide to let the ball hit the ground. The ball leaves your hand with an initial velocity of 18 feet per second when it is 5 feet above the ground. Find the highest point of the tennis ball.

• Review problems to supplement practice test;

• Pg. 287, 22, 25, 27, 29, 34, 37• Pg. 289, 39