Vectors and Parametric Equations Chapter 8 SEC 6 Vectors and Parametric Equations

Vector Equations A vector equation and equations known as parametric equations gives us a way to track the position of a moving object for given a moment of time. Tracking an airplanes movement.

Parametric Equations If a line passes through the point P1 and P2 and is parallel to the vector , the vector is also parallel to Thus must be a scalar multiple of Using the scalar t, we get Notice both are vectors. This is called the vector equation of a line. Since is parallel to the line, it is called a directional vector. The scalar t is called a parameter.

Example 1 Write a vector equation describing a line passing through P1(1, 4) and parallel to Let the line l through P1 (1, 4) be parallel to For any point P (x, y) on l, Since P1 P is on l and is parallel to for some value t. By substitution we have A vector equation can be used to describe the coordinate for a point on a line for any value of the parameter t.

Parametric Vectors For when t = 4 we get Then x – 1 = 12 and y – 4 = –8 to find ordered pair (13, –4). When t = 0 we get (1, 4) , t often represents time (Hence the t) An object moving along will be at (1, 4) at time t = 0 and point (13, –4) at time t = 4.

Parametric Vectors As we saw, the vector equation can be written as two equations relating to the horizontal and vertical components. x – x1 = ta1 and y – y1 = ta2 x = x1 + ta1 y = y1 + ta2

Example 2 Find the parametric equation for a line parallel to and passing through the point (–2, –4). Then make a table of values and graph the line. Use the general form of the parametric equations of a line with x = x1 + ta1 y = y1 + ta2 x = – 2 + 6t y = – 4 – 3t t x y – 1 – 8 – 2 – 4 1 4 – 7 2 10 – 10 t x y – 1 – 8 – 2 – 4 1 2 t x y – 1 1 2 t x y – 1 – 8 1 2

Example 3 Write a parametric equation of y = – 4x + 7. In the equation x is the independent variable and y is the dependant variable. In parametric equations t is the independent variable and x and y are dependant. So, we set the independent variables x and t equal, then we can write two parametric equations in terms of t. x = t and y = – 4t + 7 If we make a table of values and plot both the parametric and linear equations we will see they describe the same line.

Example 4 Write an equation in slope-intercept form of the line whose parametric equations are x = – 2 + t and y = 4 – 3t. 1st - Solve for t. x = – 2 + t x + 2 = t y = 4 – 3t y – 4 = – 3t

Modeling Motion Using Parametric Equations Chapter 8 Sec 7 Modeling Motion Using Parametric Equations

Objects in Motion Object that are launched, like a football, are called projectiles. The path of a projectile is called its trajectory. The horizontal distance is its range. Physicists describe the motion in terms of its position, velocity and acceleration. All can be represented by vectors.

Objects in Motion A punted kicks a ball and the initial trajectory is described by: The magnitude and direction θ will describe a vector with components . As the ball moves gravity acts on the vertical direction, while horizontal is unaffected . If we discount air friction the horizontal speed is constant throughout flight.

Objects in Motion Note the vertical component is large and positive in the beginning. Decreasing to zero at the top At the end the vertical speed is the same magnitude but the opposite direction. Parametric equations can represent the position of the ball relative to the starting point in terms of time.

Objects in Motion Write the horizontal and vertical components of the initial velocity.

Example 1 Find the initial horizontal velocity and vertical velocity of a ball kicked with an initial velocity of 18 feet per second at an angle of 37°with the ground.

Displacement due to gravity It’s about time Vertical velocity is affected by gravity, so we must adjust the vertical component by subtracting the vertical displacement of gravity we can determine the vertical position after t seconds . The equation for gravity of a free falling object is Vertical distance Displacement due to gravity time Vertical Velocity

It’s about time Because the horizontal velocity is unaffected by gravity, the horizontal position can be found after t seconds by the following. Horizontal distance Horizontal velocity time

Example 2 Sammy Baugh of the Washington Redskins has the record for the highest average punting record for a lifetime average of 45.16 yards. Suppose that he kicked the ball with an initial velocity of 26 yards per second at an angle of 72°. How far has the ball traveled horizontally and what is its vertical height at the end of 3 seconds. Because g = 32 ft/sec2 26yds = 78 ft

Example 2 Sammy Baugh of the Washington Redskins has the record for the highest average punting record for a lifetime average of 45.16 yards. Suppose that he kicked the ball with an initial velocity of 26 yards per second at an angle of 72°. How far has the ball traveled horizontally and what is its vertical height at the end of 3 seconds. Suppose that the kick returner lets the ball hit the ground instead of catching it. What is the hang time? The height as the ball hits the ground is 0. So we need to find the time, t when the height or y = 0.

Initial height Parametric equations describe objects launched from the ground. If objects are launched from above ground you must add in the initial height to the vertical component, y.

Example 3 A baseball is thrown at an angle of 5.1° with the horizontal at a speed of 85 mile per hour. The distance from the pitcher to home is 60.5 feet. If the ball is released 2.9 feet above the ground, how far above the ground is the ball when it crosses home plate? Remember to convert 85 mph to ft/sec…

Daily Assignment Chapter 8 Sections 6 & 7 Text Book Pg 524 Pg 531 #13 – 31Odd; Pg 531 #9 – 15 Odd;