Finding Arc Length when given velocity and launch angle

[kansascity816]

Homework Statement

Find the arc length of the projectile from launch until the time it hits the ground, given that
0 V is 100 feet/sec and is 45 degrees.

Homework Equations

Arc Length= ∫_a^b▒√(█(1+(f^' (x) )^2@)) dx
Arc Length of Curve= ∫_a^b▒〖v(t)dt=∫_a^b▒√((dx/dt)^2+(dy/dt) )〗^2 dt

i know these aren't pretty versions of the arc length equations, sorry about that.

The Attempt at a Solution

i'm not really sure how to go about plugging in what i was given into the equation. need help ASAP. thank you

 

[Dick]

I'd suggest using the sqrt(x'(t)^2+y'(t)^2)*dt form. What are x'(t) and y'(t) and what are the limits on t? You'll get an integral you can use a trig (or hyperbolic trig) substitution on. Just to warn you, this isn't a terribly easy problem.

 

[PtPixel]

Hey,

This reply might be a bit late but could help others who are looking for information on this kind of problem. I am a chemical engineering student working on a project for which I am designing a laminar nozzle. I needed to find the break up length (arc length to which the jet begins to break up - laminar to turbulent) of the jet of water fired from the nozzle from the angle of the initial jet, the velocity of the jet and the horizontal length of the break up point from the nozzle outlet. I have devised the following equation for finding this arc length from projectile motion physics:

X = jet break up point/point along arc we wish to find the arc length to from origin
BL = horizontal break up length/x axis displacement of point X
θ = jet angle
u = jet velocity

so initially t_BL = BL/u gives the time at which the jet breaks up (time to arc point X)
From projectile motion arc length can be shown as:

L = ∫√(u_x² + u_y²) dt from t=0 to t=t_BL (or any time chosen)
L = ∫√((ucosθ)² + (usinθ - gt)²) dt

To make things easier:
a = ucosθ
z = usinθ - gt
Giving:

L = 1/g ∫√(a² + z²) dz
1/g since dt = 1/g*dz

converting the integral boundaries:
when t = 0; z = -usinθ
when t = t_BL:
S_x = utcosθ → BL = ut_BLcosθ (x-axis displacement of point X)
t_BL = BL/(ucosθ)
∴ z = gBL/(ucosθ) - usinθ

Simplifying the boundaries:
when t = 0; -usinθ = -gt_BL - z = b
when t = t_BL; g*BL/(ucosθ) - usinθ = gBL/a - gt_BL - z = c

Finally once the arc length equation is integrated:
L = 1/g[0.5*z*√(a²+z²) + 0.5*a²* ln(z + √(a²+z²))]_b^c
(b & c being the integral boundaries)

Once b & c are substituted into the equation in place of "z", L will be the length of the arc to point X along the parabola.

Hope this helps!

 

[PtPixel]

Here's a screen shot from excel of the equation if the above text is hard to follow:

(in there there should be c² under the square root which I now noticed I missed)

Keep in mind this is to find the length to a point X, there are far easier ways to find the total parabolic arc length (from origin(0,0) until y = 0 again)

 

[Nickie]

I would approach this problem by first understanding the physical concepts involved. In this case, we are dealing with projectile motion, where an object is launched at an angle and follows a curved path due to the force of gravity. The arc length is the distance along this curved path from the launch point to the point where the object hits the ground.

To find the arc length, we can use the equation for arc length of a curve, which takes into account both the horizontal and vertical components of velocity. In this case, we are given the initial velocity (V) and launch angle (45 degrees). From this, we can find the horizontal velocity (Vx) and vertical velocity (Vy) using trigonometric equations.

Once we have Vx and Vy, we can plug them into the equation for arc length of a curve and integrate from the launch point (a) to the point where the object hits the ground (b). This will give us the total arc length traveled by the object.

It is important to note that this equation assumes a constant velocity throughout the trajectory. In reality, the velocity of the object will change due to the force of gravity, so this calculation will only give an approximation of the true arc length.

Overall, the key to solving this problem is understanding the physical concepts involved and using the appropriate equations to calculate the arc length.