Problem 13 - University Physics with Modern Physics

Question

Genesis Crash. On September \(8,2004,\) the Genesis spacecraft crashed in the Utah desert because its parachute did not open. The 210 -kg capsule hit the ground at 311 \(\mathrm{km} / \mathrm{h}\) and penetrated the soil to a depth of 81.0 \(\mathrm{cm}\) . (a) Assuming it to be constant, what was its acceleration (in \(\mathrm{m} / \mathrm{s}^{2}\) and in \(g^{\prime} \mathrm{s} )\) during the crash? \right. (b) What force did the ground exert on the capsule during the crash? Express the force in newtons and as a multiple of the capsule's weight. (c) For how long did this force last?

Step-by-Step Solution

Answer

(a) -4606.79 m/s² or -469.5 g's. (b) 967426 N or 469.5 times weight. (c) 0.0187 s.

Step 1: Convert Speed to Meters per Second

First, we need to convert the speed from kilometers per hour to meters per second. Given speed is 311 km/h. We use the conversion factor: \(1 \text{ km/h} = \frac{1}{3.6} \text{ m/s}\).\[ v = 311 \times \frac{1}{3.6} = 86.39 \text{ m/s} \]

Step 2: Calculate Deceleration Using Kinematic Equation

Use the kinematic equation \( v^2 = u^2 + 2a s \) where \( v \) is the final velocity (0 m/s, since it stops), \( u \) is the initial velocity (86.39 m/s), \( s \) is the distance (0.81 m, converted from cm), and \( a \) is the acceleration.Solve for \( a \):\[ 0 = (86.39)^2 + 2a(0.81) \]\[ a = -\frac{(86.39)^2}{2 \times 0.81} \]\[ a \approx -4606.79 \, \text{m/s}^2 \]

Step 3: Convert Acceleration to g's

The acceleration in terms of \( g \) (where \( g = 9.81 \, \text{m/s}^2 \)) is calculated as:\[ a_{g} = \frac{-4606.79}{9.81} \approx -469.5 \, g \]

Step 4: Calculate Force Exerted by Ground

Use Newton's second law, \( F = ma \), to find the force exerted by the ground, where \( m = 210 \, \text{kg} \) and \( a = -4606.79 \, \text{m/s}^2 \).\[ F = 210 \times 4606.79 = 967425.9 \, \text{N} \]

Step 5: Express Force as a Multiple of Weight

The weight of the capsule is \( mg = 210 \times 9.81 = 2060.1 \, \text{N} \).The force as a multiple of the weight is:\[ \text{Force multiple} = \frac{967425.9}{2060.1} \approx 469.5 \]

Step 6: Calculate Duration of Force

The time \( t \) during which the force acts can be found using \( v = u + at \) (with final velocity \( v = 0 \) m/s and initial velocity \( u = 86.39 \, \text{m/s}\)).Solve for \( t \):\[ 0 = 86.39 + (-4606.79)t \]\[ t = \frac{86.39}{4606.79} \approx 0.0187 \, \text{s} \]

Key Concepts

Deceleration

Deceleration is an essential concept in kinematics, particularly for understanding how objects slow down. In physics, acceleration is any change in velocity. When an object decreases its velocity, we call it deceleration. Usually, deceleration carries a negative value because the velocity is reducing. In the case of the Genesis crash, converting the capsule's impact speed helps in addressing the kinematic relations.
To calculate deceleration, you utilize the kinematic equation:

• \( v^2 = u^2 + 2as \)
• Here, \( v \) is final velocity (0 m/s since it stops), \( u \) is initial velocity, \( s \) is the penetration distance into the soil, and \( a \) is the deceleration.
• This formula rearranges to solve for \( a \), showcasing how rapidly the velocity drops to zero upon impact.
• Thus, calculating \( a \) reveals it to be approximately -4606.79 m/s\(^2\).

In reports of deceleration, it's often translated into terms of \( g \), the gravitational acceleration (9.81 m/s\(^2\)), as a more relatable measure. The deceleration calculated translates to \(-469.5\, g\), signaling a force much stronger than gravity, demonstrating the severity of the impact upon contact with the ground.

Newton's Second Law

Newton's Second Law is a fundamental principle in physics that relates to force and motion. The law states: "The force acting on an object is equal to the mass of that object times its acceleration" or mathematically, \( F = ma \). This principle is key to understanding how different masses and accelerations affect the movement and force experienced by an object.
In practical terms, this means if you know the mass of an object and its acceleration, you can calculate the force exerted on it. During the Genesis crash,

• The spacecraft had a mass of 210 kg, and when the deceleration was calculated as \(-4606.79\, \text{m/s}^2\), Newton's second law allowed for determining the force applied to it.
• The exerted force, therefore, was \( F = 210 \times 4606.79 = 967425.9 \text{ N} \).

This immense force value illustrates how high accelerations, especially when negative (as in deceleration), can lead to large forces. It highlights the significant effects quick stops have, such as those experienced in collisions or impacts.

Force Calculation

Force calculation in physics is deeply intertwined with Newton's Second Law, aiming to quantify the push or pull exerted on an object. Beyond computing force as \( F = ma \), comparing this force to other quantities can provide meaningful context.
Once the force exerted during the Genesis crash was computed,

• it became possible to express this force as a multiple of the spacecraft's weight.
• The weight of the capsule is the gravitational force on it, \( mg = 210 \times 9.81 = 2060.1 \text{ N} \).
• Comparatively, the calculated force (967425.9 N) is immensely larger, about 469.5 times its weight.
• This comparison vividly emphasizes the extraordinary forces certain situations can impart, like crash landings, where the interaction with other surfaces massively exceeds normal gravitational force experiences.

Understanding force calculation and comparing it to weight helps highlight the stresses objects endure during rapid movements or stops, key to designing safety features in vehicles or spacecraft.