Distance covered by projectile

It so happened that since I secured full marks in a test of projectile motion, my teacher challenged me to find the answer to a question. Here it is:-

$\color{blue}{\text{What is the distance covered by a projectile thrown with initial velocity}} \ \color{red}{u} \ \color{blue}{\text{and an angle of}} \ \color{red}{\theta} \ \color{blue}{\text{with the horizontal?}}$

The first thing that came to my mind was the formula $\dfrac{u^2 \sin 2\theta}{g}$ , but I realized that that was the formula for finding the horizontal range(displacement) by the projectile and not the distance.

Here is my approach for the problem:-

We break the trajectory into infinitely small bits such that it forms a right triangle of sides $dy$ and $dx$ . Let the hypotenuse be $dl$ .
Using Pythagoras' theorem:-
$\begin{aligned} dl & = \sqrt{{(dx)}^2 + {(dy)}^2}\\ \\ dl & = \sqrt{{(dx)}^2\left(1 + {\left(\dfrac{dy}{dx}\right)}^2\right)}\\ \\ \text{Integrating on both sides}:-\\ \\ \int_{0}^{l} 1 \ dl & = \int_{0}^{R} \sqrt{\left(1 + {\left(\dfrac{dy}{dx}\right)}^2\right)} \ dx \end{aligned}$

Note that $R$ here is the horizontal range of the projectile.

But my method seems too long. Is there any shorter method? Please post a shorter solution if there is one. (Air resistance is neglected).

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Comments

https://brilliant.org/discussions/thread/arc-length-of-projectile-2/

I think u might be looking for this...

abc xyz - 4 years, 11 months ago

Thanks!!!! For sharing, tht is of cool help.

Ashish Menon - 4 years, 11 months ago

i was supposed to find the same but was not asked by the teacher, i thought of it on my own and my answer is right but just one factor needed a little bit correction, thanks anyways........

A Former Brilliant Member - 4 years, 10 months ago

Nice thought then! :P

Ashish Menon - 4 years, 10 months ago

pleased to read that

I guess u are asked to find the range of the projectile since it is in horizontal distance. I suggest you make use of R=u^2sin2@/g

Ayanlaja Adebola - 4 years, 11 months ago

Nah, teacher explained the question first. Its not the horizontal range, its the length of the trajectory.

This is the only method.

However, here's a slight simplification: Substitute $x$ and $y$ in terms of the parameter $t$ .

A Former Brilliant Member - 4 years, 11 months ago

Hmm, ok i tried an alternative method too by using the formula of arc length of parabola, but that came out too tedious than this.

@Hung Woei Neoh @Swapnil Das @Rishabh Tiwari please do comment.

If I remember correctly the arc length of a curve is calculated this way

Hung Woei Neoh - 4 years, 11 months ago

Yes it is, but i tried womething with eccentricity and something "stupid" :P it was a total mess.

@Ashish Siva also see this: https://www.physics.harvard.edu/uploads/files/undergrad/probweek/sol71.pdf

Harry Jones - 4 years, 5 months ago

Thanks for the tip!

Ashish Menon - 4 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$