91Ó°ÊÓ

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 -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).

Step by step solution

01

Calculate the average acceleration (a) for stiff-legged landing

First, we need to use the formula for acceleration: \[ a = \frac{v_f - v_i}{t} \]where \( v_f = 0 \) (final velocity), \( v_i = 6.4 \ \text{m/s} \) (initial velocity), and \( t = 2.0 \times 10^{-3} \ \text{s} \).Substitute the values:\[ a = \frac{0 - 6.4}{2.0 \times 10^{-3}} = \frac{-6.4}{0.002} = -3200 \ \text{m/s}^2 \]

02

Calculate the average net force (a) for stiff-legged landing

Now, use Newton's second law to find the net force:\[ F_{\text{net}} = m \cdot a \]where \( m = 75 \ \text{kg} \).Substitute the values:\[ F_{\text{net}} = 75 \cdot (-3200) = -240,000 \ \text{N} \]The negative sign indicates the force direction is opposite to the initial movement (upward).

03

Calculate the average acceleration (b) for knee-bend landing

Again, use the formula for acceleration for bending knees:\[ a = \frac{v_f - v_i}{t} \]where \( v_f = 0 \), \( v_i = 6.4 \ \text{m/s} \), and \( t = 0.10 \ \text{s} \).Substitute the values:\[ a = \frac{0 - 6.4}{0.10} = \frac{-6.4}{0.10} = -64 \ \text{m/s}^2 \]

04

Calculate the average net force (b) for knee-bend landing

Use Newton's second law again:\[ F_{\text{net}} = m \cdot a \]where \( m = 75 \ \text{kg} \).Substitute the values:\[ F_{\text{net}} = 75 \cdot (-64) = -4800 \ \text{N} \]

05

Calculate the force of the ground (a) during stiff-legged landing

Consider both gravitational force and the net force. The gravitational force is \( F_{\text{gravity}} = mg \), where \( g = 9.8 \ \text{m/s}^2 \).Calculate:\[ F_{\text{gravity}} = 75 \cdot 9.8 = 735 \ \text{N} \]Net force from earlier was \(-240,000 \ \text{N}\). The force of the ground (upward) will be:\[ F_{\text{ground}} = F_{\text{net}} + F_{\text{gravity}} = 240,000 + 735 = 240,735 \ \text{N} \]

06

Calculate the force of the ground (b) during knee-bend landing

Use the same calculation method as in Step 5.Given \( F_{\text{gravity}} = 735 \ \text{N} \) and the net force from bending knees is \(-4800 \ \text{N}\). Now, calculate the ground force:\[ F_{\text{ground}} = F_{\text{net}} + F_{\text{gravity}} = 4800 + 735 = 5535 \ \text{N} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Acceleration

When discussing impact forces, average acceleration is a crucial concept. It helps us understand how quickly an object's velocity changes over time. This concept is essential when comparing different types of landings, such as a stiff-legged landing versus a knee-bend landing.

In mechanics, average acceleration (\( a \)) can be determined using the formula:

• \[ a = \frac{v_f - v_i}{t} \]

where:

• \( v_f \) is the final velocity,
• \( v_i \) is the initial velocity,
• and \( t \) is the time it takes for the velocity to change.

In a stiff-legged landing, the change in velocity occurs rapidly over a shorter time, resulting in a higher average acceleration. Conversely, bending the knees increases the impact duration, leading to a lower average acceleration. This difference helps in reducing the impact force experienced by the body.

Newton's Second Law

Newton's second law is pivotal in understanding how forces operate. The law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration, represented by the equation:

• \[ F_{\text{net}} = m \cdot a \]

where:

• \( F_{\text{net}} \) is the net force,
• \( m \) is the mass,
• and \( a \) is the acceleration.

This law allows us to calculate the net force acting on the person during both types of landings. With a high acceleration in a stiff-legged landing, the net force is much larger, which is why injury risk is also higher.

However, bending the knees reduces acceleration and therefore the average net force, making the landing force easier to manage and reducing the risk of injury.

Net Force

Net force is the total force an object experiences, taking into account all individual forces acting on it. In a landing scenario, the net force accounts for both the gravitational force pulling down and the force of the ground pushing up. Hence, it determines the impact experienced by the body.

The net force can be calculated using Newton's second law. When comparing a stiff-legged landing to one with bent knees, the time over which the impact occurs influences the net force greatly. With a rapid deceleration in a stiff-legged landing, the net force is substantially higher, which causes greater strain.

Reducing this force through a longer deceleration period, as in a knee-bend landing, significantly lessens the strain on the body and can help prevent injuries.

Gravitational Force

Gravitational force is a constant downward force acting on all objects with mass on Earth. It is what keeps us grounded. The force is calculated as the product of mass and gravitational acceleration:

• \[ F_{\text{gravity}} = m \cdot g \]

where:

• \( m \) is the mass of the object,
• and \( g \) is the acceleration due to gravity, approximately \( 9.8 \ \text{m/s}^2 \) on Earth.

When analyzing a landing, the gravitational force doesn't change, but the difference in how we land changes the resultant forces involved. This constant downward force is counteracted by the ground force, yet fast deceleration can make the ground force required much higher in stiff-legged landings compared to knee-bend landings. Understanding gravitational force's role helps further explain the dynamics of impact and the importance of controlled landings.