On September 8, 2004, the Genesis spacecraft crashed in the Utah desert because its parachute did not open. The 210210-kg capsule hit the ground at 311311 km/h and penetrated the soil to a depth of 81.081.0 cm. What was its acceleration (in m/s2 and in g's), assumed to be constant, during the crash?

Ch 05: Applying Newton's Laws

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 05: Applying Newton's Laws

Problem 13a

Chapter 5, Problem 13a

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 $311$ km/h and penetrated the soil to a depth of $81.0$ cm. What was its acceleration (in m/s² and in g's), assumed to be constant, during the crash?

Verified step by step guidance

Step 1: Convert the initial velocity of the capsule from km/h to m/s. Use the conversion factor: 1 km/h = 0.27778 m/s. Multiply 311 km/h by 0.27778 to express the velocity in m/s.

Step 2: Note that the final velocity of the capsule is 0 m/s since it comes to rest after penetrating the soil. The depth of penetration (displacement) is given as 81.0 cm, which should be converted to meters by dividing by 100.

Step 3: Use the kinematic equation to find the acceleration: $ v^2 = u^2 + 2as $, where $ v $ is the final velocity, $ u $ is the initial velocity, $ a $ is the acceleration, and $ s $ is the displacement. Rearrange the equation to solve for $ a $: $ a = \frac{v^2 - u^2}{2s} $. Substitute the values for $ v $, $ u $, and $ s $.

Step 4: Once the acceleration in $ \text{m/s}^2 $ is calculated, convert it to g's by dividing the acceleration by the standard acceleration due to gravity ($ g = 9.8 \ \text{m/s}^2 $). This gives the acceleration in terms of g's.

Step 5: Summarize the results by expressing the acceleration both in $ \text{m/s}^2 $ and in g's, ensuring the units are clearly stated.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Acceleration

Acceleration is the rate of change of velocity of an object over time. It is a vector quantity, meaning it has both magnitude and direction. In this context, we can calculate the acceleration of the Genesis spacecraft during its crash by using the change in velocity and the time taken for the crash. The formula for acceleration is a = (final velocity - initial velocity) / time.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration. In this problem, kinematic equations can be used to relate the initial and final velocities of the capsule, the distance it penetrated into the ground, and the time of the crash to find the acceleration.

Gravitational Force

Gravitational force is the attractive force between two masses, which on Earth gives objects weight. It is commonly expressed in terms of 'g', where 1 g is approximately 9.81 m/s². When calculating acceleration in this problem, it is useful to express the result in terms of g's to understand how the capsule's deceleration compares to the force of gravity, providing context for the severity of the crash.

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Textbook Question

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