Question

A man of mass \(83 \mathrm{~kg}\) (weight \(180 \mathrm{lb}\) ) jumps down to a concrete patio from a window ledge only \(0.48 \mathrm{~m}(=1.6 \mathrm{ft})\) above the ground. He neglects to bend his knees on landing, so that his motion is arrested in a distance of about \(2.2 \mathrm{~cm}(=0.87\) in.). ( \(a\) ) What is the average acceleration of the man from the time his feet first touch the patio to the time he is brought fully to rest? ( \(b\) ) With what average force does this jump jar his bone structure?

Short answer

The average acceleration experienced by the man during landing is about 441 m/s^2 or 45 g's. The average force exerted on his bone structure is about 36507 N or 8196 lbs.

Step-by-step solution

Calculate Final Velocity

Before the man lands, he is in free fall. His initial velocity (u) is 0 m/s. The acceleration (a) he experiences is due to gravity, and we can use the standard value of 9.8 m/s^2. The displacement (s) is the height from which he jumps, 0.48 m. Insert these values into the equation v^2 = u^2 + 2as to find the final velocity (v).

Calculate Average Acceleration of Landing

Now calculate the average deceleration or negative acceleration when the man lands. The initial velocity for this part is the final velocity from step 1 (u), the final velocity is 0 (v) as he stops, and the displacement (s) is the stopping distance of 0.022 m. Substitute these into the equation v^2 = u^2 + 2as and solve for a.

Calculate Average Force

Find the force exerted on the man's bone structure. Use the equation f=ma. The mass (m) is given as 83 kg and the acceleration (a) is the value found in step 2.

Convert the answers

Finally, convert the obtained values of acceleration and force into the customary units such as g's for acceleration and pounds (lb) for force. Remember that 1g equals to 9.8 m/s^2 and 1 N equals to 0.2248 lbs.

Key concepts

Force Calculation

Force calculation is a crucial aspect in understanding the mechanics of motion and how forces impact objects. In this scenario, when a man lands, he experiences a force upon stopping. This force can be understood using Newton's second law of motion, which states that force is the product of mass and acceleration. This is mathematically expressed as:

\[ F = ma \]
Where:

• \( F \) is the force in Newtons (N).
• \( m \) is the mass of the object, measured in kilograms (kg).
• \( a \) is the acceleration, measured in meters per second squared (m/s²).

When the man lands, his mass is given as 83 kg, and we have calculated acceleration during the impact. By substituting these values into the formula, we find the force exerted on the man's bone structure.

This force impacts how we feel after a jump or fall. The greater the force, the higher the risk of injury, which highlights the importance of forces in analyzing motion, especially in sports and physical activities.

Acceleration

Acceleration refers to the rate at which an object's velocity changes with time. In the context of this exercise, it involves two main stages. First, during the man's free-fall, and secondly, the rapid deceleration as he lands.

When falling, gravity is the only force acting on him, causing an acceleration of approximately 9.8 \( m/s^2 \). Upon landing, the objective is to compute the acceleration (actually a deceleration since he's stopping) from the moment his feet touch the ground to when he stops completely. This is key to calculating how abruptly the motion stops.

Using the formula: \[ v^2 = u^2 + 2as \]
We can calculate this acceleration:

• \( u \) is the initial velocity right before impact, found from free fall calculation.
• \( v \) is the final velocity (0 m/s) as he comes to a stop.
• \( s \) is the distance over which the stopping occurs, which is given as 0.022 m.
• \( a \) is what we solve for using this equation.

This calculated acceleration is significant as it gives insight into the forces at play over a tiny distance.

Newton's Second Law

Newton's Second Law of Motion forms the foundation for connecting the concepts of force and acceleration. According to this law, the force acting on an object is equal to the mass of that object multiplied by its acceleration. The equation is:\[ F = ma \]
This principle is essential in calculating how much force impacts an object or person, depending on their mass and how fast they're accelerating or decelerating.

In our exercise, we have a man with a mass of 83 kg and a stopping deceleration that we calculated. With these dynamics, we used Newton’s second law to find the average force exerted on the man during his landing. This force tells us how much impact his body absorbs when he drops and stops abruptly.

In practical terms:

• Understanding this law helps design better safety gear by estimating the forces acting during impacts.
• It assists engineers in calculating stress and designing structures that withstand forces.
• It emphasizes the need for techniques (like bending knees) to distribute forces and reduce impact.

Newton's law provides a logical way to predict the effects of forces in various contexts, from physics problems to real-world applications.