Simulating Motion withSimulating Motion withParametric EquationsParametric Equations

Sec. 6.3b is getting interesting…Sec. 6.3b is getting interesting…

We’ll start right in with an example…

Morris the math mole digs along a horizontal line with thecoordinate of his position (in feet) given by

Use parametric equations and a grapher to simulate his motion.Estimate the times when Morris changes direction.

3 20.1 20 110 85s t t t 0 12t

3 21 10.1 20 110 85 , 5,0 12x t t t y t

First, let’s look at:

(an arbitrary number)(an arbitrary number)

We’ll start right in with an example…

Morris the math mole digs along a horizontal line with thecoordinate of his position (in feet) given by

Use parametric equations and a grapher to simulate his motion.Estimate the times when Morris changes direction.

3 22 20.1 20 110 85 , ,0 12x t t t y t t

Next, let’s look at:

3 20.1 20 110 85s t t t 0 12t

More Practice Problems…

A rugby ball is kicked from a spot 2 feet above the groundstraight up with an initial velocity of 46 ft/sec. Graph the ball’sheight against time, find the height of the ball at 1, 2, and 3 sec,and calculate how long the ball is in the air.

20 016y t v t y

The general equation for vertical position of a projectile:

Initial velocityInitial velocity Initial heightInitial height

More Practice Problems…

A rugby ball is kicked from a spot 2 feet above the groundstraight up with an initial velocity of 46 ft/sec. Graph the ball’sheight against time, find the height of the ball at 1, 2, and 3 sec,and calculate how long the ball is in the air.

21 13.5, 16 46 2x y t t

Try graphing these:

22 2, 16 46 2x t y t t

y(1) = 32 ft, y(2) = 30 ft, y(3) = – 4 fty(1) = 32 ft, y(2) = 30 ft, y(3) = – 4 ft

The ball is in the air for approximately 2.918 sec.The ball is in the air for approximately 2.918 sec.

Huh?Huh?

More Practice Problems…

p.530-531: 38 – Capture the Flagp.530-531: 38 – Capture the Flag

1 10 0.1x t 1 3y

2 100 9x t 2 3y

(a) Check the graph window [0,100] by [–1,10]

Use simultaneous mode, and note that it’s the processof graphing that’s important, not the final graph…

(b) Who capture the flag, and by how many feet?

The faster runner captures the flag when the slowerrunner is still 4.1 feet away from the flag.

More Practice Problems…

p.531: 40 – Height of a Pop-upp.531: 40 – Height of a Pop-up

20 016y t v t s

(a) Write an equation that models the height of the ball as afunction of time t.

216 80 5t t (b) Use parametric mode to simulate the pop-up.

1 6x Graph and trace:

21 16 80 5y t t 0 6t

(c) Use parametric mode to graph the height against time.

2x tGraph and trace:

22 16 80 5y t t 0 6t

More Practice Problems…

p.531: 40 – Height of a Pop-upp.531: 40 – Height of a Pop-up

24 16 4 80 4 5y

(d) How high is the ball after 4 sec?

69Solve both graphically, and algebraically:

The ball is 69 ft above the ground after 4 sec.The ball is 69 ft above the ground after 4 sec.

(e) What is the maximum height of the ball? How many secondsdoes it take to reach its maximum height?

Solve graphically…

When When tt = 2.5 sec, the ball is at it maximum height = 2.5 sec, the ball is at it maximum heightof 105 ft.of 105 ft.

Projectile Motion with Parametric

EquationsOur last new stuff in Sec. 6.3

To this point, we’ve talked about motion in only onedirection (i.e., along only one axis)…

Now, imagine a baseball thrown from a point y feetabove ground level with an initial speed of v ft/sec atan angle 0 with the horizontal:

0

0

x

y

y0

v0

0v sin 00

v cos 00

What is the componentWhat is the componentform of the initial velocity?form of the initial velocity?

v = v cos 0, v sin 00 0 0

Now, imagine a baseball thrown from a point y feetabove ground level with an initial speed of v ft/sec atan angle 0 with the horizontal:

0

0

x

y

y0

v0

0v sin 00

v cos 00

The The horizontal horizontal andandvertical vertical components ofcomponents of

an object’s motion in thisan object’s motion in thissituation are independentsituation are independentof each other, and can beof each other, and can bemodeled by the followingmodeled by the following

parametric equations:parametric equations:

0 cosθx v t 20 016 sin θy t v t y

Our first example:

Kevin hits a baseball at 3 ft above the ground with an initialspeed of 150 ft/sec at an angle of 18 with the horizontal. Willthe ball clear a 20-ft wall that is 400 ft away?

Equations modeling the path of the ball:

150cos18x t 216 150sin18 3y t t

When with the ball reach the wall? After 2.804 sec.After 2.804 sec.

What is the height of the ball at this point? y = 7.178 ft.y = 7.178 ft.

Can we use our calculators to solve this???Can we use our calculators to solve this???

More Practice…

Proctor kicks for points in a rugby game. He place-kicks theball with an initial speed of 63 ft/sec at an angle of 46 withthe horizontal. If the ball heads directly towards the 10 ft-highcrossbar that is 116 ft from Proctor, will the ball clear thecrossbar?

63cos 46x t 216 63sin 46y t t The equations:

116 63cos 46 t Time it takes the ball to reach the crossbar:

2.651sect

More Practice…

Proctor kicks for points in a rugby game. He place-kicks theball with an initial speed of 63 ft/sec at an angle of 46 withthe horizontal. If the ball heads directly towards the 10 ft-highcrossbar that is 116 ft from Proctor, will the ball clear thecrossbar?

216 2.651 63sin 46 2.651y

Height of the ball at this time:

7.710fty No, the ball will fall short by 2.290 ft.No, the ball will fall short by 2.290 ft.

Can we solve Can we solve graphicallygraphically??

More Practice…

Starting with the same situation as the previous example,there is now a 6 ft/sec split-second wind gust just as Proctorkicks the ball. If the wind acts in the horizontal directionbehind the ball, will the kick clear the crossbar?

6 63cos 46x t 216 63sin 46y t t

Yes, the ball will clear the crossbar by 8.699 ft.Yes, the ball will clear the crossbar by 8.699 ft.

The new equations:

More Practice…

Tony and Sue are launching yard darts 20 ft from the front edgeof a circular target of radius 18 in. on the ground. If Sue throwsthe dart directly at the target, and releases it 4 ft above theground with an initial velocity of 25 ft/sec at a 55 angle, will thedart hit the target?

25cos55x t 216 25sin 55 4y t t The equations:

The dart lands when y = 0… Find the time when this happens:

1.452sect

More Practice…

Tony and Sue are launching yard darts 20 ft from the front edgeof a circular target of radius 18 in. on the ground. If Sue throwsthe dart directly at the target, and releases it 4 ft above theground with an initial velocity of 25 ft/sec at a 55 angle, will thedart hit the target?

25cos55 1.452x 20.822ftNow, find how far the dart traveled in this time:

Yes, the dart lands about 10 inchesYes, the dart lands about 10 inchespast the front edge of the targetpast the front edge of the target

Refer to this example when you do #47 in the HW!!!Refer to this example when you do #47 in the HW!!!

So, will the dart land in the target?