The distance (not horizontal nor vertical) of a rock traveled in a projectile

[yuki_meiseki]

Homework Statement

A small rock of mass m is kicked from the edge of a perfectly vertical cliff of height h, giving
it an initial velocity ~vo that’s purely horizontal. The rock lands at a distance d away from
the bottom of the cliff. The ground there is assumed horizontal and we neglect air resistance.
Derive the distance traveled by the rock in terms of h and d only.

Homework Equations

(h) y=-gt^2/2
(d) x=v0t
Using the two parametric equations above and the arc length formula for parametric equations.
Integrate[Sqrt[(y')^2+(x')^2],t,0,x/v0]

The Attempt at a Solution

We could not solve the integral
and the professor did not allow us to have vo and t in the final answer!

Help...it's due tmr...

 

[rl.bhat]

Distance traveled = Integration(v*dt) = Int[sqrt{(gt)^2 + vo^2}]*dt
s = Int[vo*sqrt{(gt/vo)^2 + 1}]*dt = vo*Int.[sqrt{(gt/vo)^2 + 1}]*dt
Let gt/vo = 2*1/2*gt^2/vo*t = 2h/d = tan(theta)
(g/vo)*dt = sec^2(theta)*d(theta) and dt = [sec^2(theta)*d(theta)]vo/g. Substitute in s, we get
s = (vo^2/g)*Int[sec^3(theta)*d(theta). If you solve this by integration by parts, we get
s = (d^2/4h)*[sqrt{1 + 4h^2/d^2}*2h/d + log(sqrt{1 + 4h^2/d^2}+2h/d )]

 

[robphy]

Distance traveled = Integration(y*dx) = Int(gt^2/2)*dx
x=v0t. Hence t = x/vo. Put it in the above integration.
Integration(y*dx) = Int(gt^2/2)*dx = Int(g*x^2/2vo^2)*dx = g*x^3/6vo^2.
Put vo^2 = x^2/t^2 in g*x^3/6vo^2. It becomes g*t^2*x^3/6x^2.But g*t^2 = 2y
So the distance traveled = 2yx/6 = hd/3

So the distance traveled has units of length^2 ?

It seems that the OP is looking for the arc-length along a segment of a parabola.
Can you write the integrand in simplest form?

 

[rl.bhat]

Distance traveled = Integration(v*dt) = Int[sqrt{(gt)^2 + vo^2}]*dt
s = Int[vo*sqrt{(gt/vo)^2 + 1}]*dt = vo*Int.[sqrt{(gt/vo)^2 + 1}]*dt
Let gt/vo = 2*1/2*gt^2/vo*t = 2h/d = tan(theta)
(g/vo)*dt = sec^2(theta)*d(theta) and dt = [sec^2(theta)*d(theta)]vo/g. Substitute in s, we get
s = (vo^2/g)*Int[sec^3(theta)*d(theta). If you solve this by integration by parts, we get
s = (d^2/4h)*[sqrt{1 + 4h^2/d^2}*2h/d + log(sqrt{1 + 4h^2/d^2}+2h/d )]