The free-fall acceleration on a large asteroid, in the vacuum of space, is 0.15 m/s². A spacecraft hovering 500 m above the surface drops a 25 kg payload wrapped in a padded jacket. What is the payload's impact speed?

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Identify the known values: The free-fall acceleration \( g = 0.15 \; \text{m/s}^2 \), the initial height \( h_0 = 500 \; \text{m} \), and the initial velocity \( v_0 = 0 \; \text{m/s} \) since the payload is dropped from rest.

Use the kinematic equation \( v^2 = v_0^2 + 2gh \) to find the final velocity \( v \) of the payload just before impact. Here, \( v_0 = 0 \), \( g = 0.15 \; \text{m/s}^2 \), and \( h = 500 \; \text{m} \).

Substitute the known values into the equation: \( v^2 = 0 + 2(0.15)(500) \). Simplify the expression to find \( v^2 \).

Take the square root of both sides to solve for \( v \): \( v = \sqrt{2(0.15)(500)} \).

Interpret the result: The value of \( v \) represents the impact speed of the payload just before it hits the surface of the asteroid.

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주요 개념

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Free-Fall Acceleration

Free-fall acceleration refers to the acceleration of an object due solely to the force of gravity acting on it, without any other forces like air resistance. On the asteroid mentioned, this acceleration is 0.15 m/s², which is significantly lower than Earth's 9.81 m/s². This means that objects will fall more slowly on the asteroid, affecting their impact speed.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. In this scenario, we can use the equation v² = u² + 2as, where v is the final velocity, u is the initial velocity (0 m/s for a dropped object), a is the acceleration (0.15 m/s²), and s is the distance fallen (500 m). This equation allows us to calculate the impact speed of the payload.

Impact Speed

Impact speed is the velocity of an object just before it collides with another object or surface. It is influenced by the height from which the object is dropped and the acceleration due to gravity. In this case, calculating the impact speed of the payload involves determining how fast it is moving just before it hits the surface of the asteroid after falling 500 m.