# How to find angle of a curve in xy coordinate plane?

1. Aug 14, 2003

### Matt Jacques

Hi,

I think it is atan of the derivative of the curve, is that right?

Thanks

Matt

2. Aug 14, 2003

### mathman

The derivative is the slope of the tangent line. The atan is thus the angle of the tangent line with the x axis.

3. Aug 14, 2003

### Matt Jacques

Thanks for the quick reply. Yeah, that is what I thought. However, I have a followup question.

For the vertical position of a projectile the formula is:

y = initialheight + VyT - .5(9.8)T^2

The derivative of that is

d/dx = Vy - 9.8T

I did a sample problem, Vi=4, angle = 30º, initial height = 1.2

Total Time: .739 seconds
Height of Max time: .202 seconds
Distance: 2.56 m

if I take atan( Vy - 9.8T) when t = 0 the angle comes up as 63º, not 30º which it should be.

Hmm?

Last edited: Aug 14, 2003
4. Aug 15, 2003

I started to work out the problem, but you didn't state it quite clearly....

Do you mean V0x is the initial Vi in the direction of vector i ?

Last edited: Aug 15, 2003
5. Aug 15, 2003

### HallsofIvy

Staff Emeritus
dy/dx is the arctan of the angle the curve makes with the horizontal.

dy/dt has no geometric meaning (in an xy-coordinate system).

If Vi= 4 is the initial speed and the projectile is fired at a 30 degree angle to the horizontal, then the initial vertical speed is Vy= 4 cos(30)= 4(1/2)= 2 and the initial horizontal speed is
Vx= 4 sin(30)= 4([sqrt](3)/2)= 2 [sqrt](3).

The vertical speed would be given by dy/dt= 2- 9.8t and the horizontal speed (assuming gravity is the only force) by
dx/dt= 2[sqrt]3. At t= 0, dy/dt= 2 and dx/dt= 2[sqrt](3).
By the chain rule, dy/dx= (dy/dt)/(dx/dt)= 2/2[sqrt](3)= 1/[sqrt](3)= [sqrt](3)/3, the tangent of 30 degrees.

6. Aug 15, 2003

### Matt Jacques

Thanks, Hallsofivy. You created a good thorough reply. However, I think you accidentally swapped cosine and sine in Vx and Vy.

Anyway, for the benifit of searches, here is the formula for the angle:

Last edited: Aug 15, 2003
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9823197722434998, "perplexity": 4164.330505619274}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541518.17/warc/CC-MAIN-20161202170901-00451-ip-10-31-129-80.ec2.internal.warc.gz"}