Parametric Equations: Motion in a Plane Notes for Section 6.3 In Laman s terms: Parametric equations allow us to put and into terms of a single variable known as the parameter. Time, t, is a common parameter used in this section. We can define and values each with its own equation in terms of t along with a specified interval for t values. Formal Definition: The graph of the ordered pairs, where f () t and g() t are functions defined on an interval I (of t -values) is a parametric curve. f () t and g() t The variable t is the parameter. I is the parameter interval. are parametric equations for the curve. Note: If no t interval is indicated, we assume t, or t When we give parametric equations and a parameter interval, I, for a curve, we have parametrized the curve. Therefore, parametrization of a curve consists of not onl the parametric equations but also the interval of t -values. Eample : a) Find a parametrization of the line AB through points A,3 and B4,6 b) What is the t-interval that will generate the segment AB? a)first, find the equation for the line through A and B in slope intercept form.. becomes m, so 3 ( ) becomes 3 3 and finall 4 If we assign t as the parameter and equate it with, we can then write a set of parametric equations. = t & = 4 t b) Letting our parameter interval be t,4 segment. or t 4, we can generate the endpoints of the t t t 4 - - 3 4 4 6 A B,3 4,6
Eample : Parametrize the circle with center (-,-4) and radius. We know rcos and rsin defines an point on a circle, so and rcost h rsint k is the parametric form for a circle. cost and sint 4, t 0, parametricall defines this circle. Eample 3: a) Graph b hand the parametric equations t and 3t, where t 3. t - 0 3 t - - - 7 3t -3 0 3 6 9 b) Now let us graph the parametric equations on the calculator on the same interval: T T T 3 Setting the window: T Tmin: - X-min: - Y-min: -5 T ma: 3 X-ma: 0 Y-ma: 0 Tstep: 0. X-scl: Y-scl: c) Finall, let s use the calculator to graph the same equations on the following t intervals. i) t 3, ii) t,3 Eample 4:.5 feet awa from the side of a 40 foot tower, a rock is dropped. The vertical path of the falling rock can be represented b.5. The height of the rock as it freefalls can be represented b 6t 40. Let t represent seconds, where t 0,5. See if ou can set our window to accommodate this situation. Use our table to find the height of the rock, 3 and 5 seconds after it is dropped. second 404 ft. 3 seconds 76 ft. 5 seconds 0 ft.
Eliminating the Parameter Sometimes we need to eliminate the parameter from a set of parametric equations and be able to identif the function in its Cartesian (Rectangular) form. This means getting rid of the parameter, t, and obtaining a single equation in terms of onl and. Remember: If no t interval is indicated, we assume Here are some eamples to practice. t, or t Eample 5: Eliminate the parameter and identif the graph of the parametric curves. a) t & t where t,4 Solve for t in the first equation and substitute for t in the second equation. st equation t t t or t nd equation t 3 3 is a line segment in slope intercept form from 5,4 to7,. b) t & 3t nd equation solved for t : t Now substitute it for t in the st equation, and And now solving for... 3 t becomes: 3 ) 9 9 9 9( 3 This is a parabola which opens to the right and has verte,0.
3 cos t & sin t where t 0, Hint: c) cos sin t t h k r 4cos t 4sin t If cos t & sin t then, t t 4(cos sin ) 4() 4 0 0 This is 3 4 of a circle from 3 0, with radius and center 0,0. Eample 6: Two opposing plaers in Capture the Flag are 00 ft apart. On a signal, the run to capture a flag that is on the ground midwa between them. The faster runner, however, hesitates for 0. sec. The following parametric equations model the race to the flag: =0(t 0.), = 3 = 00 9t, = 3 a. Simulate the game in a [0,00] b [-,0] viewing window with t starting at 0. Graph simultaneousl. b. Who captures the flag and b how man feet? 0( t0.) 50 00 9t 50 t0. 5 9t 50 t 5.sec. t5.556 sec. Faster runner still wins b 0.456 seconds. Second runner is still 4. feet from the flag at t = 5. seconds.
The equation for vertical projectile motion in terms of time: 6t v0t s0. Eample 7: A baseball is hit straight up from a height of 5 ft with an initial velocit of 80 ft/sec. a. Write an equation that models the height of the all as a function of time t. t t 6 80 5 b. Use a parametric mode to simulate the pop-up. t t 3 & 6 80 5 c. Use parametric mode to graph the height against time. [Hint: Let (t) = t] t t t & 6 80 5 -min: 0 -min: -0 -ma: 0 -ma: 0 d. How high is the ball after 4 sec? 69 ft. 6(4) 80(4) 5 e. What is the maimum height of the ball? How man seconds does it take to reach its maimum height? b b Use verte formula., f a a (.5 sec, 05ft.) at.5 seconds, the ball 05 feet high. Projectile Motion when looking at horizontal and vertical components. v t and t v t s ( o cos ) 6 ( o sin ) o If a projectile is launched at an initial height of s o feet above the ground at an angle of from horizontal and the initial velocit is vo feet per second, the path of the projectile is modeled b the parametric equations shown above.
Eample 8: A baseball plaer is at bat and hits a ball at a height of 4 feet. The ball leaves the bat at 0 ft/sec towards the center field fence, which is 45 feet awa. The fence is feet high. If the ball leaves the bat at an angle of elevation of 39, will the ball be a homerun? Equations for the ball: t and t t Equations for the fence: Window on Calculator: (0cos39 ) 6 (0sin39 ) 4 45 and t T-min: 0 X-min: -0 Y-min: 0 T-ma: X-ma: 500 Y-ma: 00 T-step: 0. X-scl: 50 Y-scl: 0 Trace to get an -value close to 45 ft. Looking at the table: Table Setup TblStart = 4.55 Tbl =.00 At t= 4.558, the ball is 45.07 feet awa and 5.807 feet high, so it is a homerun! Sinusoidal Problems Using Parametric Equations: r cos h & r sin k Eample 9: Kristin is riding on a Ferris wheel that has a radius of 30 ft. The wheel is turning counterclockwise at a rate of one revolution ever 0 seconds. Assume the lowest point of the Ferris wheel (6 o clock) is 0 feet above the ground and that Kristin is at a point marked A (3 o clock) at time t=0. Use parametric equations to find Kristin s position seconds into the ride. 30' 0' r = radius = 30 feet Now we need as a function of time. revolution radians / sec. so, t 0 sec. 0 5 5 Thus, 30cos t 0 & 30sin t 40 5 5 When t, Kristin' s position is 9.7 ft., 68.53 ft. This means she is 9.7 feet to the right of the ale, and 68.53 feet above ground.