A designer is creating an obstacle for an obstacle course where a person starts on a moveable platform of height H from the ground. The person grabs a rope of length L and swings downward. At the instant the rope is vertical, the person lets go of the rope and attempts to reach the far side of a water-filled moat. The left side of the moat is directly below the position where the person will let go of the rope. The designer runs several tests in which the rope has different lengths and moves the platform such that the rope is always initially horizontal. The designer notices that the person cannot land on the other side if the length L is very short. The designer also notices that the person also cannot land on the other side if the length L is very close to the height H.

Assume the size of the person is much smaller than the lengths L and H. Let D represent the horizontal distance from below the release point to where the person lands.

Required:
a. Why does the person land in the moat if the rope's length is very short?
b. Why does the person land in the moat if the length is nearly the same as the height of the platform?

in this case we see that the speed to cross the well grows a lot (it goes towards infinity) therefore we do not have enough speed in the movement to cross

* when L → 0 very short string

the speed of the platform is very small, so we do not have the minimum required value

vox = √ (g / (2 (H)) D

Explanation:

For this exercise we are going to solve it using conservation of energy to find the velocity of the body and the launch of projectiles to find the velocity to cross the well.

Let's start with the projectile launch

as the body leaves the vertical its velocity must be horizontal

x = v₀ₓ t

y = y₀ + [tex]v_{oy}[/tex] t - ½ g t²

when reaching the ground its height of zero (y = 0) and the initial vertical velocity is zero

t = √ 2 y₀ / g

we substitute

x = vox √2y₀ / g

v₀ₓ = √(g / 2y₀) x

In the exercise, it tells us that the width of the well is D (x = D) and the initial height is the height of the platform minus the length of the rope (I = H - L)

v₀ₓ = √(g /(2 (H -L)) D

this is the minimum speed to cross the well.

Now let's use conservation of energy

starting point. On the platform

[tex]Em_{o}[/tex] = U = m g H

final point. At the bottom of the swing

Em_{f} = K + U = 1 / 2m v² + m g (H -L)

as there is no friction the mechanical energy is conserved

Em_{o} = Em_{f}

m g H = 1 / 2m v² + m g (H -L)

v = √ (2gL)

let's write our two equations

the minimum speed to cross the well

v₀ₓ = √ (g /(2 (H -L)) D

the speed at the bottom of the oscillatory motion

v = √ (2g L)

we analyze the extreme cases

* when L → H chord too long

in this case we see that the speed to cross the well grows a lot (it goes towards infinity) therefore we do not have enough speed in the movement to cross

* when L → 0 very short string

the speed of the platform is very small, so we do not have the minimum required value

vox = √ (g / (2 (H)) D

From this analysis we see that there is a range of lengths that allows us to have the necessary speeds to cross the well

V₀ₓ = v

g / (2 (H -L) D² = 2g L

4 L (H- L) = D²

4 H L - 4 L2 - D² = 0

L² - H L - D² / 4 = 0

let's solve the quadratic equation

L = [H ± √ (H2-D2)] / 2

we assume that H> D

L = ½ H [1 + - RA (1 - (D / H) 2)]

The two values of La give the range of values for which the two speeds are equal

batolisis batolisis · Answer 2 · 2022-01-13T14:42:22+01:00

A) The person lands in the moat if the rope's length is very short because :

The speed of the platform is less than the required minimum speed

B) The person lands in the moat if the rope length is similar to the height of the platform because :

The speed required to cross the moat approaches infinity

Following the assumptions;

size of the person is much smaller than L and H

D = horizontal distance

The conditions that will cause the person to land on the moat

The person will land in the moat when the rope's length is very short because as the rope reduces in length the speed reduces as well such that the speed of the platform goes below the required minimum speed which will enable the person cross over. while

As the magnitude of the length tends towards the same magnitude of the height the speed required to cross the moat increases towards infinity and this speed cannot be attained by the person hence he will land in the moat.

Hence we can conclude that The person lands in the moat if the rope's length is very short because The speed of the platform is less than the required minimum speed and The person lands in the moat if the rope length is similar to the height of the platform because,the speed required to cross the moat approaches infinity.

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