Math Problem Statement

Answer this

If air resistance is ignored, on which planet would a space probe with a mass of 250 kg have the highest speed after falling 25 m?

Solution

To solve this problem, we need to determine on which planet the space probe will reach the highest speed after falling 25 meters, assuming no air resistance. The speed of an object falling from a certain height can be calculated using the following kinematic equation:

v=2ghv = \sqrt{2 g h}

Where:

  • vv is the final speed,
  • gg is the acceleration due to gravity on the planet,
  • hh is the height the object falls from.

Since the question asks which planet provides the highest speed, we can ignore the mass of the probe because the equation does not depend on mass. Therefore, the key factor is the gravitational acceleration gg.

Now, from the table, we compare the values of gg for each planet:

  • Venus: g=8.9m/s2g = 8.9 \, \text{m/s}^2
  • Earth: g=9.8m/s2g = 9.8 \, \text{m/s}^2
  • Uranus: g=8.7m/s2g = 8.7 \, \text{m/s}^2
  • Saturn: g=9.0m/s2g = 9.0 \, \text{m/s}^2

Clearly, Earth has the highest acceleration due to gravity at 9.8m/s29.8 \, \text{m/s}^2. Since higher gg leads to a higher final speed, the space probe will achieve the highest speed on Earth.

Thus, the correct answer is:

D. Earth


Would you like further details or have any questions?

Here are 5 related questions:

  1. How does gravitational acceleration vary with planetary mass?
  2. What other factors, besides gravity, could affect the speed of a falling object on a planet?
  3. How would the presence of an atmosphere impact the falling speed of an object?
  4. Why is mass irrelevant when calculating the speed of an object in free fall in this case?
  5. How would the height from which an object falls affect its final speed?

Tip: Gravitational acceleration is stronger on planets with more mass or higher density, but surface gravity depends on both mass and the radius of the planet.

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Math Problem Analysis

Mathematical Concepts

Physics
Kinematics
Gravitational Acceleration

Formulas

v = sqrt(2gh)

Theorems

Kinematic equations under gravity

Suitable Grade Level

Grades 9-12