When jumping straight down, you can be seriously injured if...

When jumping straight down, you can be seriously injured if you land stiff-legged. One way to avoid injury is to bend your knees upon landing to reduce the force of the impact. A $75-\mathrm{kg}$ man just before contact with the ground has a speed of $6.4 \mathrm{~m} / \mathrm{s}$. (a) In a stiff-legged landing he comes to a halt in $2.0 \mathrm{~ms}$. Find the average net force that acts on him during this time. (b) When he bends his knees, he comes to a halt in $0.10 \mathrm{~s}$. Find the average net force now. (c) During the landing, the force of the ground on the man points upward, while the force due to gravity points downward. The average net force acting on the man includes both of these forces. Taking into account the directions of the forces, find the force of the ground on the man in parts (a) and (b).

When jumping straight down, you can be seriously injured if you land stiff-legged. One way to avoid injury is to bend your knees upon landing to reduce the force of the impact. A $75-\mathrm{kg}$ man just before contact with the ground has a speed of $6.4 \mathrm{~m} / \mathrm{s}$. (a) In a stiff-legged landing he comes to a halt in $2.0 \mathrm{~ms}$. Find the average net force that acts on him during this time. (b) When he bends his knees, he comes to a halt in $0.10 \mathrm{~s}$. Find the average net force now.
(c) During the landing, the force of the ground on the man points upward, while the force due to gravity points downward. The average net force acting on the man includes both of these forces. Taking into account the directions of the forces, find the force of the ground on the man in parts (a) and (b).

Key Concepts

Impulse-Momentum Theorem

This concept relates the impulse applied to an object to the change in its momentum. It shows that the force applied over a given time interval (impulse) is equal to the change in the object's momentum. Understanding this relationship is key when analyzing collisions or impacts, as it helps in calculating the average force during rapid deceleration.

Newton’s Second Law

Newton’s Second Law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. This fundamental principle is used to understand how different forces affect an object’s motion and to calculate forces during changes in velocity, such as during impacts or deceleration events.

Impact Force and Deceleration

This concept involves the relationship between the rate of deceleration and the magnitude of the force experienced during an impact. By spreading the change in momentum over a longer time interval, the force exerted on the object is reduced. This is a critical idea in safety practices, where extending the duration of an impact (e.g., by bending knees) can significantly lower the injurious forces.

Free Fall and External Forces

In free fall situations, both gravitational force and the reaction force from the ground (normal force) act on the object. Analyzing the vector sum of these forces, while considering their directions, is essential to accurately evaluate the net force during landing or collision events. This helps in understanding how external forces combine to produce the resulting motion or deceleration.

Transcript

00:01 So we've got a 75 kilogram man jumping onto the ground.

00:04 And just before he hits the ground, he's got a speed of 6 .4 meters per second.

00:08 So the problem says that when you jump and you land stiff -legged, meaning you don't bend your legs or your knees, you can really hurt yourself.

00:17 But you can avoid injury by bending your knees when you land.

00:21 And that reduces the force.

00:22 So why does that reduce the force? well, we're going to look at this in terms of momentum.

00:26 And what was the man's momentum right before he hit the ground? and what was his momentum after hitting the ground? so when we talk about momentum, we put a prime for after a collision and no prime for before a collision.

00:38 So we've got this 75 kilogram man.

00:40 He, before the collision, has a downward velocity of 6 .4 meters per second.

00:45 After the collision, he has no velocity.

00:47 We're going to use positive direction to be down.

00:50 We're going to figure out he lands stiff -legged in 2 milliseconds.

00:54 What's the average net force acting on him? he bends his legs.

00:58 He comes to a stop in 0 .1 seconds.

01:00 What's the average net force on him then? then we're going to try to figure out, well, that's net force.

01:05 Part of that net force is his downward gravitational force.

01:09 So how much of that net force is the ground going up is the final question for a and b.

01:15 So let's go ahead and figure all this out.

01:18 So we can calculate the change in momentum, but to relate that then to force takes a little bit of a connection.

01:25 Well, we know that the impulse of a force on an object is equal to that object's change in momentum.

01:30 But we also know that the impulse of a force on an object is the average value of that net force times the amount of time it acts on the object.

01:40 Let's put in net here because it's a net force acting.

01:42 So if this is equal to impulse and this is equal to impulse, that means these two things are equal to each other.

01:48 So we can now say that the change in momentum of the man is equal to the net force acting on the man times the amount of time that it acts on him.

01:59 So we want to solve for the net force.

02:01 Well, that just means that the net force is going to be his change in momentum over the amount of time for which that net force is acting on him.

02:11 So let's go ahead and expand this so it makes a little bit of sense.

02:16 So change in momentum is p prime, momentum after the collision, minus p, momentum before the collision, divided by delta t now.

02:27 So momentum after is the mass of the object times its velocity after the collision minus the mass of the object times its velocity before the collision.

02:36 And then that's going to be divided by time...

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