Question

Hi everyone ! I just watched the 4th video and I wondered what's the distance that the projectile travels (i.e the perimeter of the parabola), I predicted it's h+OS but I'm not sure.. Please somebody can think about it with me ? :)

Answer 1

the exact value of this distance is d=\[\int\limits_{0}^{ts}v(t)dt\] but I couldn't figure out how to calculate it !

Answer 2

the exact answer is
\[\int\limits_{0}^{T} speed[t] d(t)\]
this will give u answer

Answer 3

In general if you have a function \[f : \mathbb{R} \rightarrow \mathbb{R}\] which is differentiable (or at least piece wise differentiable, then the distance of the graph of f between x = a and x = b is given by \[\int\limits_{a}^{b} \sqrt{1 + [f'(x)]^{2}} dx\]
It's not hard to convince yourself of this formula as the length of an infinitesimal line element is \[\sqrt{dx^2 + dy^2} = \sqrt{1 + (dy/dx)^2 } dx = \sqrt{1 + f'(x)^2} dx\]

So for the parabola of motion is equivalent to calculating this integral. Suppose the parabola has equation\[f(t) = - g t^2/2 + v_0t + h_0, \hbox{then} f'(t) = - g t + v_0\] and the integrand of the above equation is\[\int\limits_{0}^{T} \sqrt{1 + ( g t - v_0)^2} dt\]

Now you need to know T and evaluate this integral using a substitution\[u = \sqrt{1 + ( g t-v_0)^2}\]

This looks a little scary but it's actually not so bad and it's a great calculus exercise, so I won't do it unless you're really stuck.

Answer 4

rightly this is what I'm wanting to calculate, I made a physical approach (decomposing the velocity into x component and a y component) ending up with the same integral. Thank you for the maths explanation, it made it clearer for me ! Now I have to evaluate this integral \[\int\limits_{0}^{T}\sqrt{1+t^{2}}dt\] (I removed constant terms for simplicity), doing the subtitution you suggested, now I need to evaluate this : \[\int\limits_{0}^{\sqrt{u ^{2}-1}}(u ^{2}\div \sqrt{u ^{2}-1})du\]
which still I cannot solve..
P.S I mistyped the prediction : it's 2h+OS, thinking that the projectile goes up and down besides the distance OS.

Answer 5

No. In your first integral substitute t = sinh(u). If you don't know the function hyperbolic sin, sinh, look it up. It's very helpful.