Motion of a Food Pack on a Rope

In summary, the speed of the food pack is given by x(x^{2}+h^{2})^{-1/2}V^{1}_{boy} and the acceleration is h^{2}(x^{2}+h^{2})^{-3/2}V^{2}_{boy} as the boy walks away from the vertical rope with constant velocity V^{1}_{boy}, holding the free end of the rope with his hand. The Pythagorean theorem is used to calculate the distance covered by the pack and the differentiation of this theorem gives the instantaneous velocity and acceleration.
  • #1
Whatazarian
3
0
[SOLVED] Motion in one dimension

Homework Statement


To protect his food from hungry bears, a boy scout raises his food pack with a rope that is thrown over a tree limb at height h above his hands. He walks away from the vertical rope with constant velocity V[tex]^{1}_{boy}[/tex], holding the free end of the rope with his hand.

(a) Show that the speed V of the food pack is given by x(x[tex]^{2}[/tex]+h[tex]^{2}[/tex])[tex]^{-1/2}[/tex]V[tex]^{1}_{boy}[/tex] where x is the distance he has walked away from the vertical rope.

(b) Show that the acceleration a of the food pack is h[tex]^{2}[/tex](x[tex]^{2}[/tex]+h[tex]^{2}[/tex])[tex]^{-3/2}[/tex]V[tex]^{2}_{boy}[/tex].

Homework Equations



h[tex]^{2}[/tex](x[tex]^{2}[/tex]+h[tex]^{2}[/tex])[tex]^{-3/2}[/tex]V[tex]^{2}_{boy}[/tex]=a
x(x[tex]^{2}[/tex]+h[tex]^{2}[/tex])[tex]^{-1/2}[/tex]V[tex]^{1}_{boy}[/tex]=V

The Attempt at a Solution



First of all I would advise you all to look at the attached diagram of the situation (as expressed in the question).

First, I concluded that V[tex]^{1}_{boy}[/tex]= x/t, since the boy travels a distance of x in some time t. Then, knowing that the boy started at x = 0, I calculated the distance the pack covered in this time would be (x[tex]^{2}[/tex]+h[tex]^{2}[/tex])[tex]^{1/2}[/tex] by Pythagoras' Theorem. Hence, V= ((x[tex]^{2}[/tex]+h[tex]^{2}[/tex])[tex]^{1/2}[/tex])/t, and since t =x/V[tex]^{1}_{boy}[/tex] we can conclude that V = V[tex]^{1}_{boy}[/tex]((x[tex]^{2}[/tex]+h[tex]^{2}[/tex])[tex]^{1/2}[/tex])/x by substitution.

Am I right or wrong? Obviously if you switch the distances around, the equation would come out as expected, but, I cannot fathom why and how they would be the other way around.

b) Taking V from a), and dividing by t to give acceleration, a = V[tex]^{2}_{boy}[/tex]((x[tex]^{2}[/tex]+h[tex]^{2}[/tex])[tex]^{1/2}[/tex])/x[tex]^{2}[/tex]
or a = (V[tex]^{2}_{boy}[/tex]x)/((x[tex]^{2}[/tex]+h[tex]^{2}[/tex]))

From there I don't know what to do since the acceleration is changing, so my method is probably wrong.

Any help would be appreciated.
 
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  • #4
Whatazarian said:

Homework Statement


To protect his food from hungry bears, a boy scout raises his food pack with a rope that is thrown over a tree limb at height h above his hands. He walks away from the vertical rope with constant velocity V[tex]^{1}_{boy}[/tex], holding the free end of the rope with his hand.

(a) Show that the speed V of the food pack is given by x(x[tex]^{2}[/tex]+h[tex]^{2}[/tex])[tex]^{-1/2}[/tex]V[tex]^{1}_{boy}[/tex] where x is the distance he has walked away from the vertical rope.

(b) Show that the acceleration a of the food pack is h[tex]^{2}[/tex](x[tex]^{2}[/tex]+h[tex]^{2}[/tex])[tex]^{-3/2}[/tex]V[tex]^{2}_{boy}[/tex].

Homework Equations



h[tex]^{2}[/tex](x[tex]^{2}[/tex]+h[tex]^{2}[/tex])[tex]^{-3/2}[/tex]V[tex]^{2}_{boy}[/tex]=a
x(x[tex]^{2}[/tex]+h[tex]^{2}[/tex])[tex]^{-1/2}[/tex]V[tex]^{1}_{boy}[/tex]=V

The Attempt at a Solution



First of all I would advise you all to look at the attached diagram of the situation (as expressed in the question).

First, I concluded that V[tex]^{1}_{boy}[/tex]= x/t, since the boy travels a distance of x in some time t. Then, knowing that the boy started at x = 0, I calculated the distance the pack covered in this time would be (x[tex]^{2}[/tex]+h[tex]^{2}[/tex])[tex]^{1/2}[/tex] by Pythagoras' Theorem. Hence, V= ((x[tex]^{2}[/tex]+h[tex]^{2}[/tex])[tex]^{1/2}[/tex])/t, and since t =x/V[tex]^{1}_{boy}[/tex] we can conclude that V = V[tex]^{1}_{boy}[/tex]((x[tex]^{2}[/tex]+h[tex]^{2}[/tex])[tex]^{1/2}[/tex])/x by substitution.

Am I right or wrong? Obviously if you switch the distances around, the equation would come out as expected, but, I cannot fathom why and how they would be the other way around.

I think the formula in the problem statement is correct. You are right to assume that the pack is moving at the same rate as the rope is, but the rope is not moving at the same rate as the boy. The only time this would be true is if the rope and the boy were in a straight line (as if he were dragging it along the ground).

If I could make a suggestion: [tex] dx/dt = v_{boy} [/tex] and you can get the Pythagorean theorem states that [tex] l^2 = x^2 + h^2 [/tex] where l is the length of the hypotenuse. Perhaps you would consider differentiating the Pythagorean theorem as a first step? This would also give you dl/dt, which is the rate of change of the hypotenuse, or, in other words, the speed of the rope.

I couldn't open your attachment, but I'm assuming the rope makes a right triangle, with the food pack on the perpendicular to the direction of the boy.

Shelly
 
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  • #5
SheldonG said:
I think the formula in the problem statement is correct. [tex] (x^2+h^2)^{1/2}[/tex] is the length of the hypotenuse, but the food pack is not traveling along the hypotenuse, it is traveling along h. And what is wanted is the instantaneous velocity, not the average velocity, which x/t would give you.

If I could make a suggestion: [tex] dx/dt = v_{boy} [/tex] and the Pythagorean theorem states that [tex] l^2 = x^2 + h^2 [/tex] where l is the length of the hypotenuse. Perhaps you would consider differentiating the Pythagorean theorem as a first step?

I couldn't open your attachment, but I'm assuming the rope makes a right triangle, with the food pack on the perpendicular to the direction of the boy.

Shelly

I think you are correct, thank you. I was under the misconception that It was an average velocity, but, when I treat it as an instantaneous velocity the answer falls through easily and also makes more sense.

I can now also solve part b) with the new information.

Cheers :D
 

Related to Motion of a Food Pack on a Rope

1. What is "motion in one dimension"?

"Motion in one dimension" refers to the movement of an object along a straight line, without any changes in direction.

2. What are some examples of motion in one dimension?

Examples of motion in one dimension include a car driving in a straight line, a ball rolling down a ramp, and a person walking in a straight line.

3. How is motion in one dimension represented?

Motion in one dimension is often represented using a position-time graph, which shows the object's position at different points in time.

4. What is the difference between speed and velocity in one dimension?

Speed refers to the rate at which an object is moving, while velocity includes both the speed and the direction of motion in one dimension.

5. How is the acceleration of an object in one dimension calculated?

The acceleration of an object in one dimension can be calculated by dividing the change in its velocity by the change in time, or by finding the slope of its velocity-time graph.

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