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Parametric Equations and Motion - Problem 3
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
One problem you might see when you're working on Parametric equations, is a problem where parametric equations are used to model projectile in motion. In this problem, Brocc Samson throws a 16 pound shot from a height of 6 feet, with an initial speed of 40 feet per second at angle of 45 degrees.
Parametric equations for the path of x equals 40 cosine 45 degrees times t, and y equals 6 plus sine 45t, minus 16t². These equations work for t greater than or equal to zero. You can imagine that he's releasing the shot at t equals zero, and then the shot travels for some amount of time and it hits the ground at some point. Now, assuming the ground is level, how far is Brocc's throw?
When you're measuring a throw, you measure horizontally, so we want to find what horizontal distance the throw covers. In order to do that, we have to find out when it hits the ground. So we're going to need the y equation here, it hits the ground when y is zero. So I have to solve this equation; 0 equals 6 plus 40 sine 45t minus 16t².
Now let's keep in mind that t is greater than, or equal to 0, so we only want answers that are greater than, or equal to 0. Now I'm going to make a little substitution, let's just call this b. It will make writing the answer a little easier. I'm going to use the quadratic formula.
Now I'm solving for t, so I get t equals -b, plus or minus the square root of b², minus 4 times -16 times 6 all over 2 times 16. So I'm going to need my calculator for this. I'll get approximate values. Now I've already stored this value as b, so I'm just going to type -b plus the square root of b², minus 4, times -16, times 6. That gives me 6.125, I divide that by -32 and I get -.19. That's one solution, t equals negative .19, of course this solution is no good to use, because we want t to be greater than or equal to 0.
So we just calculated -b plus, let's calculate -b minus. So I go back and I change the plus to minus, enter divided by -32 and I get 1.959. 1.96, that's the value we want. That's the only solution that's greater than zero, so what this means is that the shot actually comes back down, hits the ground after 1.96 seconds, so y equals zero at that point.
Now I need to find out how far the shot's gone. So I need this equation; x equals 40 cosine 45 times t, and I'm going to use that value of t I just found. So x equals 40 cosine 45, times t; times 1.96. This will be an approximation because 1.96 is an approximation.
SO again I use my calculator, and I get 55.4 feet. What that means is that when the shot actually comes down, it's travelled a horizontal distance of 55.4 feet. We have no idea how high it went, but it's gone 55 and a half feet above.
All right, one thing you might be asked to do on your homework, is to graph the parametric equations that we've just discussed on your calculator. That's an important calculator skill. Let me show you how to do that using the TI-84.
So we're looking at the TI-84. The first thing I want to do is enter these paramateric equations. So I hit y equals, and notice I've got y1 equals, y2 equals, y3 equals. I can enter functions of x here, but I can't enter parametric equations in this mode. I have to change mode, so go to the mode button, press that, and then going down, I notice that I'm in function mode here, fourth row down on the left I need to change to parametric mode, Par.
So go down to the fourth row, over 1 and hit enter and that changes us to parametric mode. While I'm at it, I'm going to change from radians to degrees for that 45 degrees. So I hit it enter, now I'm in degree mode. So let's hit the y equals button again. Now we see X1t, Y1t, X2t, Y2t I can enter parametric equations here.
Now the the first one was 40 times cosine of 45 times t, that's t. The second one was 6 plus 40 sine 45, times t, minus 16t². Now before we graph it, I want to check the window. All right, this is a pretty good window the projectile was actually in motion for two seconds. So let's change that t max to 2. I'll leave the t step at .1, x min and x max. Remember that the shot didn't quite go 60 feet, so let's have x max be 60. The x scale, that's the increment, is 10 and y min and y max, I like 40 for y max. Let's see what that graph looks like.
Here we are, so we have this is graph which is a shape roughly like a parabola, it actually is a parabola. That's one of the reasons parabolas are so important, is that they can be used to model projectile motion.
Now I just want to show you really quickly how you can find points on this graph by hitting the trace button. Hit the trace button, that's under F4. The first thing you see is T0, and X0, T equals 0, x equal 0 and y equals 6. It's giving you the point when T equals 0.
Now if I hit the right arrow, it will advance time by .1 second. That was my t step, but you can also advance it much quicker by just entering the number you want like .5 return. So now once t equals .5, x is about 14, and y is about 16. I can do that again 1 return, t equals 1 the shot was at 28 feet, x equals 28 and y equals 18, 1.5 etcetera.
Now, remember that the shot was in the air for about 1.96 seconds, so I can just type that in. Notice we have x equals 55.4, that's how far we determine that the shot went approximately. And notice that y is -.3. Now our model doesn't know we've entered the values slightly too large, y shouldn't be negative. The shot should actually stop, but the model doesn't know that. Let's just take this as approximately 0, and this is where the shot actually lands.
That's how you set a parametric equations on the TI-84. Remember to change mode to parametric from function, and if you need to change from radians to degrees., always remember we work with radians a lot in Mathematics. So remember to change back when you're done.
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