## anonymous 5 years ago curvilinear motion...

1. anonymous

parametric equations: y(t)=-4.9t^2 +4.864t+1.2192 and x(t)=5.08t... i found the equations for velocity individually etc etc. i got s(t)=-1.9291t+.9574 for the speed of the particle at any time, t. and the next question is asking to find dy/dx?

2. anonymous

You can find dy/dx using the chain rule. That is,$\frac{dy}{dx}=\frac{dy}{dt}\frac{dt}{dx}$All you need to do is take the derivatives of your equations with respect to time and multiply them appropriately.

3. anonymous

that is what i did to get the speed function s(t)... if i found dy/dx as that, why is it asking for it again, or is it in reference to another derivative of something else?

4. anonymous

I'm thinking the speed is given by,$s(t)=\sqrt{\left( \frac{dy}{dt} \right)^2+\left( \frac{dx}{dt} \right)^2}$Is that how you found it?

5. anonymous

since you're looking for the magnitude of velocity at that point.

6. anonymous

nope but that does make sense, now that i reread the question, it just wants that formula, which then means that the next question wants me to take the derivative of that?

7. anonymous

Do what I wrote for the speed, and do what I wrote for dy/dx...you should get two different answers. Does that help?

8. anonymous

what does that s(t) formula tell me in terms of the problem and what does the dy/dx tell me?

9. anonymous

im sorry but im trying to understand the formulas

10. anonymous

s(t) is the speed - the magnitude of the velocity. dy/dx just gives you the rate of change of the y co-ordinate as x varies - so it's just looking at how vertical displacement varies with horizontal...nothing special.

11. anonymous

so finding dy/dx pretty much has nothing to do with the speed function and my teacher just put it in there randomly...?

12. anonymous

Sometimes it's useful to have motion problems in terms of displacements only, excluding time. It's easier for understanding the geometrical properties of the path taken.

13. anonymous

Well, speed is defined as the *time* rate of change of distance. dy/dx makes no mention of time. I'm not sure what your teacher intends.

14. anonymous

which one excludes time?

15. anonymous

ah, but if i divide dy/dt by dx/dt i still have dy/dx in terms of t

16. anonymous

Well, sometimes we solve one coordinate for time and sub. it into the other coordinate to combine the coordinates into a relationship between each other, that's all.

17. anonymous

Yes, you may have it in terms of t, and that's fine. That would just allow you to answer the question, "What is the path rate of change in y versus x at time t?"

18. anonymous

Or you could eliminate t using one of the equations and sub. it in and you'd have dy/dx in terms of x and y only...you can do anything! Okay, not *anything*.

19. anonymous

ok, how about the trajectory of the particle at time 0 and time 1.2? i dont think i even heard about what that means

20. anonymous

The trajectory is the path a moving object follows through space as a function of time. You have your displacement vector in terms of t, namely, $(x(t),y(t))$All you'd do is sub. in the times to get a displacement vector for the object at that time.

21. anonymous

The question is just asking, "Where is the object headed at time 0 and at time 1.2s?"

22. anonymous

its asking the angle that it left my hand (time 0) and the angle that it hit the ground (time 1.2)

23. anonymous

Actually, I should have said, "Where is the object?", not 'headed', since that would be velocity.

24. anonymous

Oh - so that's why you needed dy/dx in terms of t...remember what dy/dx means geometrically? It's the gradient of the function. dy/dx is telling you where the particle is headed.

25. anonymous

$\frac{dy}{dx}(t)=\tan \theta$

26. anonymous

27. anonymous

I'm assuming from what you've said you're throwing something in this question. Draw a right-angled triangle where the hypotenuse is sloping up. Let theta be the angle...wait, I'll draw something.

28. anonymous

Look at this...

29. anonymous

so basically its the tennisball project in two directions. up and out. i threw the ball, from 1.2192m and it took 1.2s to land 6.096m away. i got all the equations above then it asked for the formula for speed of the particle at any time. and right after that said "step 6: find dy/dx."

30. anonymous

tan (theta) = (dy/dt)/(dx/dt) = dy/dx

31. anonymous

that makes sense... what do it do with that?

32. anonymous

Well, from that setup (picture), dy/dt is always vertical and dx/dt is always horizontal. Their sizes will change depending on time, and therefore, so will dy/dx. All that picture is doing is to show the relationship between the angle with the horizontal and the velocity vector. You're looking for the angle at a certain time. You have dy/dx as a function of t, and that picture shows you how dy/dx relates to that angle (i.e. tan(theta) = dy/dx). So, find dy/dx at your times and take the inverse tangent to find the angles. Check to see if the answers make sense.

33. anonymous

that makes complete sense, lemmy try that

34. anonymous

and so just making sure...: dy/dx (t) gives me both dy vertical height and dx horizontal distance as a function of time?

35. anonymous

at the same time?

36. anonymous

Nearly. What it's doing is giving you the *rate of change* of y with respect to x (as a function of time).

37. anonymous

So you're getting the rate of change in y wrt x given the time...you don't have to know anything about x (or y) explicitly (like you do when you're asked more elementary questions like, y=x^2 -> dy/dx = 2x. To fing dy/dx here, we need to know x).

38. anonymous

right, being the derivative, but yeah it does both in relevance to time... jeez im clearly too tired if im being this stupid

39. anonymous

so i got that it left at approx 137 degrees and landed at 83 degrees which looks right

40. anonymous

yes, you're right (not about being stupid!).

41. anonymous

nah man im definitely being stupid right now... idk where you are but here on the east coast its 4am and i got school first thing tomorrow

42. anonymous

alright so to get the greatest speed ill use the dy/dx formula?

43. anonymous

83 degrees...might be (180-83) degrees if you think about the fact the coordinate system and how the ball lands. I'll draw something.

44. anonymous

Well, to get the greatest speed, I would use s(t) and take the derivative wrt to t to find the time at which the speed was greatest. Then you can use that time to find position.

45. anonymous

yeah it would be about -263 or +97 degrees technically but same thing in this perspective... and i can understand why dy/dx is a speed formula but whats the whole root of dy/dt sqrd + dx/dt sqrd) ?

46. anonymous

what is wrt?

47. anonymous

wrt = with respect to

48. anonymous

whats the difference between using s(t) and dy/dx, would they get me the same speed?... would dy/dx get me "speed" ?

49. anonymous

The whole 'root' thing is because of the following. The velocity vector is$v=(x'(t),y'(t))$and since the speed is the magnitude of the velocity vector, you have$v^2=v.v=(x'(t),y'(t)).(x'(t),y'(t))=x'(t)^2+y'(t)^2$

50. anonymous

But this is the square of speed, so we have to take the square root of both sides.

51. anonymous

$v=\sqrt{x'(t)^2+y'(t)^2}$

52. anonymous

Re. using s(t) and dy/dx...speed is the time rate of change, which is what s(t) is. dy/dx is *not* a time rate of change, it's a displacement rate of change; it measures the change in the y-coordinate as a function of the change in x-coordinate. It's different. It gives you information about the geometry of what's going on - as in the case where you have to find angles for where things fall.

53. anonymous

4am...are you getting up to study, or have you not gone to bed?

54. anonymous

not gone to bed

55. anonymous

Have you got a test or something?

56. anonymous

technically i dont actually take calculus

57. anonymous

Is this just for physics or something?

58. anonymous

cant take physics either, yet... im a freshman in hs

59. anonymous

So what's this for?

60. anonymous

Torture?

61. anonymous

torture to who?

62. anonymous

You.

63. anonymous

I wasn't being rude :)

64. anonymous

haha no, this is my fun

65. anonymous

i got half an hour and about 7 speed questions that i know are wrong now

66. anonymous

so i really have no option but taking the root of that big equation? :(

67. anonymous

If you want the speed, no.

68. anonymous

first thing i want is speed at time 0

69. anonymous

dx/dt will be 0 and dy/dt will be 4.864... so i sqare that then root it? which is still 4.864...

70. anonymous

I would just take the derivative of x(t) and y(t) and sub. the times in to find numerical values, square each, add, take square root of sum (if your aim is computation only).

71. anonymous

what would be another aim? than computation?

72. anonymous

Well, if you're asked to find the actual formula...if you're not asked to, and only asked to find values (and you're in an exam), it would take up unnecessary time.

73. anonymous

$y(t)=-4.9t^2+4.864t+1.2192 \rightarrow y(0)=1.2192$$x(t)=5.08t \rightarrow x(0)=0$So,$s(0)=\sqrt{1.2192^2+0^2}=1.2192$

74. anonymous

dy/dt or y(t) ??

75. anonymous

oh

76. anonymous

yeah - I'm tired too

77. anonymous

lol you on the eat cost too?

78. anonymous

No, other side of the world. Sydney. Off sick.

79. anonymous

are you a physicist or something or just enjoy math?

80. anonymous

I'm trained in pure maths and physics. I'm a UK citizen, worked with people in physics from Oxford, Cambridge, Yale.

81. anonymous

wait a sec... it asked for the speed as the ball approached the max height.. if the max height is when y' is 0 then ill end up with dy/dt is 0 and dx/dt is 5.08?

82. anonymous

that is freaking awesome.

83. anonymous

:)

84. anonymous

that makes me wonder, how many teens have you worked with that your first answer is "torture" to math :P

85. anonymous

and i need help with this one: on what intervals is the speed increasing and decreasing? (i dont want to use an analysis -.- )

86. anonymous

lol, I've worked with people of all ages and abilities. All I care about is the attitude.

87. anonymous

You don't want to use an analysis? What do you mean?

88. anonymous

logic says that its decreasing till the max height and increasing therein after

89. anonymous

i dont want to use any derivative analysiss

90. anonymous

oh

91. anonymous

that makes no sense, my calculations say that it had a speed of approx, 3 when i let go... 8 when it hit the ground, and 5.08 at the max height

92. anonymous

One sec...

93. anonymous

oh i see my mistake... omg this is pathetically horrible ><

94. anonymous

i got 7 this time :DDD

95. anonymous

the speed should be fastest when it hits the ground, right?

96. anonymous

Well, you could find the maximum height by understanding that the vertical component of velocity must be zero there...so $\frac{dy}{dt}=0$should give you the time, which you can use to find the height, y. I know you said no calculus, but I figured you already had y'(t).

97. anonymous

98. anonymous

Yes, the speed should be fastest when it hits the ground. You get that from energy considerations.

99. anonymous

yeah i already got everything else

100. anonymous

niceee finally i can glue this crap together lol

101. anonymous

Good! I can't believe you're up at ~4am and doing this...lol.

102. anonymous

Do you want to study math/science after school?

103. anonymous

?

104. anonymous

like after i graduate, keep studying?

105. anonymous

Yeah

106. anonymous

i mean, i intend to get a Ph.D in both mathematics and probably EE... ill be doing this till the day i die

107. anonymous

XD

108. anonymous

good

109. anonymous

( when you can answer) why did you get into math?

110. anonymous

School...I accelerated my studies. After school, I had offers for medicine and law, but stuck to mathematics and physics...to my father's grief. I've worked in and biophysics with a chemist from Ox. and nuclear physicist (turned biophysicist) from Yale. Now, I'm taking a break.

111. anonymous

a break?

112. anonymous

Sabbatical.

113. anonymous

?

114. anonymous

Just a formal break from university stuff.

115. anonymous

are you a professor or a researcher?

116. anonymous

I was a researcher. I'm just teaching for a while without the research.

117. anonymous

i can tell that you come on here often haha

118. anonymous

Lol, I know...it's a nice distraction from other things I should be doing.

119. anonymous

i could probably use a little more help on a related rates problem DX

120. anonymous

i'll see what i can do

121. anonymous

whether baloon is launched on a windless day, goes straight up... rises at a rate of 1000ft/minutes, observer stands 10ft away, his eyes are 6 ft off the ground at what rate is the angle of elevation changing when the baloon is 10000 ft high

122. anonymous

i can do related rates, i just cant figure out how to set this one up ...

123. anonymous

well, you won't have, by definition, an angle of elevation until the balloon hits 6 ft. So I would consider finding the vertical velocity and other 'initial' conditions at the y=6ft mark and work from there...

124. anonymous

...whaa??

125. anonymous

one sec.

126. anonymous

no i get what youre saying... nvrmnd that "whaa?" but we are taking the calculations mostly at 10000ft where that 6ft will have been very much covered

127. anonymous

and yeah i tried that, but i still cant set up the equation right

128. anonymous

with the +6 or -6 or even without it, its the trig that i cant get down ><

129. anonymous

Okay, I'm going to draw something.

130. anonymous

actually, given that it is 6 o clock now.. its probably best if you dont... i need to get my 15 minutes of sleep XDDD

131. anonymous

oh, i'll continue and you can go...

132. anonymous

ill prolly be back tonight, i always liked helping people with their math

133. anonymous

iight im out- thank you for everything btw

134. anonymous

no probs.

135. anonymous

Had a break...so, you need to find $\frac{d \theta }{dt}$ as some function of height. From the pic. I sent, you should see that$\tan \theta = \frac{h}{10} \rightarrow \frac{d}{dt}\tan \theta = \frac{d}{dt}\frac{h}{10} \rightarrow \sec^2 \theta \frac{d \theta}{dt}=\frac{1}{10}\frac{dh}{dt}$That is,$\frac{d \theta }{dt}=\frac{dh/dt}{10\sec^2 \theta}=\frac{dh/dt}{10(1+\tan^2 \theta)}=\frac{dh/dt}{10(1+h^2/100)}$$\frac{d \theta}{dt}=\frac{10(dh/dt)}{100+ h^2}$You're given$\frac{dh}{dt}=1000ft/\min$and your height at this point is (10,000 - 6) ft since our calculations for angle of elevation begin at 6ft. Then$\frac{d \theta }{dt }|_{h=9994}=\frac{10 \times 1000}{100+(10000-6)^2}\approx 1 \times 10^{-4} radians/\min$

136. anonymous

From your question, I assume any acceleration on the balloon in the vertical is ignored.

137. anonymous

thats precisely what i got! and yes, there is no acceleration in this case

138. anonymous

[thumbs up]