Maximum Distance a Projectile Moves Up an Inclined Plane

[EnricoHendro]

Homework Statement

A projectile is fired up an incline (incline angle θ) with an initial speed V[SUB]0[/SUB] at an angle φ with respect to the horizontal (φ > θ) as shown in Figure P4.86.
(a) Show that the projectile travels a distance, d, up the incline, where :

Relevant Equations

$\displaystyle d=\dfrac{2{V_0}^2 \cos\theta \sin(\theta-\phi)}{g^2 \cos^2 \phi}$
(Edited by Mentor emeritus)

the red line is the initial velocity, the grey parabola is the path of the projectile.

hi there...I'm kinda stuck at the part b of this problem. I can do part a with no problem.
can anybody explain to me how to do the differentiation needed to solve part b?? by explain I mean explain the steps, not just showing the results. Thank you

 

[archaic]

Consider $d=f(\theta)$ and differentiate with respect to $\theta$. An example:
$$\frac{d}{d\theta}\left(a\sin(\theta)\right)=a\cos(\theta)$$
You need to find:
$$\frac{d}{d\theta}\left( \frac{2v_0^2}{g^2\cos^2(\phi)}\cos(\theta)\sin(\theta-\phi) \right)$$
Any term that does not involve $\theta$ is to be treated as a constant.
You should know what it means to maximize $d(\theta)$, right?

 

[EnricoHendro]

Consider $d=f(\theta)$ and differentiate with respect to $\theta$. An example:
$$\frac{d}{d\theta}\left(a\sin(\theta)\right)=a\cos(\theta)$$
You need to find:
$$\frac{d}{d\theta}\left( \frac{2v_0^2}{g^2\cos^2(\phi)}\cos(\theta)\sin(\theta-\phi) \right)$$
Any term that does not involve $\theta$ is to be treated as a constant.
You should know what it means to maximize $d(\theta)$, right?

hmm...what I'm stuck at is how do you differentiate the equation from part a?? I mean I can memorize the result of the differentiation of the equation from part a, but I do not know how to differentiate it.

 

[archaic]

hmm...what I'm stuck at is how do you differentiate the equation from part a?? I mean I can memorize the result of the differentiation of the equation from part a, but I do not know how to differentiate it.

Oh, are you not taking calculus along with physics? Or maybe you haven't seen how to differentiate trigonometric functions yet? If it is the latter, please see this http://tutorial.math.lamar.edu/Classes/CalcI/DiffTrigFcns.aspx
If you haven't taken calculus yet, though, then you can read the second and third chapter as ordered here http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx

 

[EnricoHendro]

Oh, are you not taking calculus along with physics? Or maybe you haven't seen how to differentiate trigonometric functions yet? If it is the latter, please see this http://tutorial.math.lamar.edu/Classes/CalcI/DiffTrigFcns.aspx

nope...I am not taking calculus along with physics...in fact I am studying physics by myself...you know, self taught...so I am not taking any calculus. thanks a lot for the link by the way

 

[archaic]

nope...I am not taking calculus along with physics...in fact I am studying physics by myself...you know, self taught...so I am not taking any calculus. thanks a lot for the link by the way

You are welcome! If you are more of a learner through videos, you can always check Khan Academy:
https://www.khanacademy.org/math/ap-calculus-bc
They also have videos on Physics and other subjects, explore the website a bit, you might like it.

 

[anorlunda]

I am studying physics by myself...you know, self taught...so I am not taking any calculus.

Mathematics is the language of physics. You'll need the calculus first. The advice from @archaic was good.

 

[Mark44]

nope...I am not taking calculus along with physics...in fact I am studying physics by myself...you know, self taught...so I am not taking any calculus. thanks a lot for the link by the way

Since you haven't taken calculus yet, then there is no way you can do problems in which you are asked to find a derivative or calculate an integral. Physics is often taught in two different tracks: one in which calculus knowledge is not required, and another that assumes the student has studied at least the first quarter or semester of calculus.

Since there is nothing more we can help you with until you get up to speed with calculus, I am closing this thread.