An 80.0-kg sky diver jumps out of a balloon at an altitude...

An $80.0-\mathrm{kg}$ sky diver jumps out of a balloon at an altitude of $1000 \mathrm{~m}$ and opens the parachute at an altitude of $200.0 \mathrm{~m}$. (a) Assuming that the total retarding force on the diver is constant at $50.0 \mathrm{~N}$ with the parachute closed and constant at $3600 \mathrm{~N}$ with the parachute open, what is the speed of the diver when he lands on the ground? (b) Do you think the sky diver will get hurt? Explain. (c) At what height should the parachute be opened so that the final speed of the sky diver when he hits the ground is $5.00 \mathrm{~m} / \mathrm{s} ?$ (d) How realistic is the assumption that the total retarding force is constant? Explain.

Key Concepts

Work?Energy Principle

This principle states that the net work done on an object is equal to its change in kinetic energy. In problems like these, it is used to relate the loss in gravitational potential energy, minus the work done by retarding (non?conservative) forces, to the final kinetic energy of the diver. It allows one to determine the speed at various stages of the motion by equating energy changes to work done against forces such as air resistance or drag.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. For falling objects, the energy lost during descent is converted into kinetic energy and is also partially used in doing work against resistive forces. Understanding this concept is crucial to model how altitude differences affect the available energy for acceleration or deceleration during free fall.

Retarding Forces and Air Resistance

Retarding forces, often modeled as drag or air resistance, act opposite to the direction of motion and reduce an object’s kinetic energy as it falls. In simplified analyses, these forces can sometimes be assumed constant over certain intervals, though in reality they typically depend on factors such as velocity, shape, and air density. Recognizing the role of these forces is key to analyzing the decelerated descent of objects like skydivers.

Parachute Dynamics

Parachute dynamics involve the sudden increase in drag force when a parachute is deployed, which significantly reduces the falling speed of a skydiver. This concept highlights how alterations in surface area and the resulting increase in air resistance can greatly impact the net work done against gravity, thereby controlling the final impact speed. It also touches on the practical considerations of when and how a parachute should be opened for safety.

Non?Conservative Forces in Energy Analysis

Non?conservative forces, such as air resistance, dissipate mechanical energy from a system via work not recoverable as mechanical energy. In problems involving falling objects, these forces must be accounted for as they alter the energy balance between gravitational potential energy and kinetic energy. This concept explains why energy conservation alone is insufficient and why work done by non?conservative forces must be included in the analysis.

Transcript

00:01 All right, so we've got a height of this, and we got a guy that's falling, and he decides to open his parachute at 200 meter mark.

00:12 Between that, there's a certain friction coefficients when the parachute isn't opened, and then there's a new friction coefficient when the parachute is opened.

00:21 So up here, we're going to have gravitational potential energy.

00:26 And down here, just before we hit the ground, we're going to have kinetic energy done.

00:31 And all between here, we're going to have the work done by when the parachute is opened, plus the work done by the parachute being closed.

00:42 And we could probably abbreviate that as wp, work done by parachute.

00:49 So that means that the speed at which the parachuter hits the ground, if they open the parachute at 200 meters, is going to be this, her initial energy of gravitational potential energy plus that kinetic energy plus the work done by parachutes and we're just going to do this subtract the work done by the parachute and that's going to equal the kinetic energy which is given by one half mv squared multiply the two divide by m we're given two times gravitational potential energy, work done by parachutes, divided by m, the square root equals v.

01:46 And then wp is going to be given by the work done by its respective coefficients.

01:53 So the 50 newtons is going to be for the first 800 meters, so 50 times 800.

01:58 And then 3 ,600 is going to be for the last 200, so we're going to do 3 ,600 times 200.

02:06 And that's going to be wp.

02:09 So then when we add those values in there, calculate ug, multiple by 2, divide by m and take the square roots.

02:16 We get that the velocity is about 26 meters per seconds, which is, sir, more like 25.

02:24 And for reference, that's about 55 miles per hour.

02:29 So he's going to go splat.

02:33 Not very safe, so he's definitely not going to be okay or even alive.

02:41 And then the next part, we're just going to ask what is the safer height to attain a speed of 5 meters per second.

02:49 So this one's going to require a little bit more thought process.

02:53 Still going to be the same equation up here, but this time we're going to include the wnwc.

02:59 And we're actually going to include their distances.

03:02 So it's going to look like this.

03:04 We still have gravity energy.

03:08 We still have kinetic energy.

03:10 But this time we've already set it to be a certain speed, so 5 meters per second.

03:17 So at 5 meters per second plus the work done by the open parachute, which is going to be, let's just substitute that.

03:30 So it's going to be 50 newtons times a distance plus 360, 3.

03:39 3 ,600 meters times a second distance.

03:49 And these two distances, d plus d2, is going to equal y.

03:59 So just a little rearranging of the equation, substituting the five meters per second.

04:04 And now we're just gonna get d in terms of y...

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