Question

the table shows the mass of and acceleration due to gravity for several planets in the solar system. 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?

planet	mass, (10^{24}) kg	acceleration due to gravity, ( \text{m/s}^2 )
earth	5.97	9.8
uranus	86.8	8.7
saturn	568	9

options:
a. uranus
b. earth
c. saturn
d. venus

Explanation:

Step1: Recall the kinematic equation

For an object in free fall (ignoring air resistance), the final velocity \( v \) after falling a distance \( d \) with initial velocity \( u = 0 \) (assuming it starts from rest) is given by the equation \( v^{2}=u^{2} + 2ad \). Since \( u = 0 \), this simplifies to \( v=\sqrt{2ad} \), where \( a \) is the acceleration due to gravity and \( d \) is the distance fallen.

Step2: Analyze the variables

The distance \( d = 25\space m \) is the same for all planets. The mass of the space probe does not affect the final velocity (from the equation \( v=\sqrt{2ad} \), mass is not a factor in free - fall velocity when air resistance is ignored). So, the final velocity depends only on the acceleration due to gravity \( a \). The larger the value of \( a \), the larger the value of \( v \) (because \( v=\sqrt{2ad} \) and \( d \) is constant).

Step3: Compare the accelerations

We have the accelerations due to gravity for each planet:

• Venus: \( a = 8.9\space m/s^{2} \)
• Earth: \( a=9.8\space m/s^{2} \)
• Uranus: \( a = 8.7\space m/s^{2} \)
• Saturn: \( a=9\space m/s^{2} \)

Comparing these values: \( 9.8>9 > 8.9>8.7 \). Earth has the largest acceleration due to gravity among the given planets.

Answer:

B. Earth