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ascofield33

Member

Registered: 2014-03-03

Posts: 4

Non-Linear Parametric motion help!!

Many people enjoy the game of darts. there are even national darts tournaments. In one dart game, you stand 8 feet from a dart board and throw three darts per round, attempting to score points totaling 500.
a.) Suppose you are most accurate when you throw a dart at an angle 20° to the horizontal. If the bull's eye is at the same height as your release point, about what initial velocity must you impart to the dart to hit the bull's eye?
b.) How long is the thrown dart in the air?
c.) suppose your dart-throwing opponent is most accurate when he throws at an initial velocity of 30 feet per second. At what angle should he throw his dart to hit the bull's eye?
d.) About how long is his dart in the air?

I was able to solve parts a and b, but I am confused on how to solve for parts c and d. I'm sure once I figure out part c I will be able to figure out part d. Any help would be greatly appreciated! Thanks!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,582

Re: Non-Linear Parametric motion help!!

hi ascofield33

There's a single formula for both parts here so I'll derive that first.

Let the initial velocity be U at angle A to the horizontal

Then horizontally, the (constant) velocity is U cos A; the distance travelled is 8 and let's call the time of flight T

D = ST will work here so

Vertically the initial velocity is U sin A; the acceleration is -g (acts to oppose the upward travel); the distance travelled is zero as the launch height and target height are the same; and the time of flight is T again.

Using

Substitute in for T and re-arrange

It's reasonable to cancel U and cosA as the problem would be silly if either were zero so

or, using the double angle formula

So, if you know A, you can calculate U, and if you know U, you can calculate A.

Take care to have g in feet/s/s.

After that getting T is easy from the first equation.

Hope that helps, 

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob