Understanding Gravity: Earth vs Moon

Gravity on earth vs on the Moon.

Planetary Body	Acceleration due to Gravity
Earth	g $\approx$9.81 m/s2
Moon	gmoon $\approx$1.62 m/s2

This means objects fall faster on earth than on the Moon!

Explanation: How Falling Motion Works

When an object is dropped

• Initial velocity 𝑢 = 0 (if simply released, not thrown).
• It accelerates downward at a constant rate because of gravity.
• The distance fallen and the speed at a moment in time follow these equations:
  
  Velocity after time t:

since 𝑢 = 0,

Distance fallen after time t:

with 𝑢 = 0,

Distance – Time graph and Velocity – Time Graph for falling objects on Earth vs the Moon.

A Distance – Time Graphs hows how the distance fallen changes with time:
– The graph is curved, not streigth.
– This curve means the object is accelerating.
– As time passes, the object covers more distance each second.
Earth vs Moon
On Earth, the curve is steeper.
On the Moon, the curve is less steep.
A steeper the line means a greather acceleration.

A Velocity- Time Graphs hows how the Velocity of a falling object changes with time:
– The graph is a straight line.
– It starts from origin because the objects starts from rest.
– A straight line means constant acceleration.
Earth vs Moon
On Earth, the line is steeper.
On the Moon, the line is less steep.
The slope (gradient) of a velocity-time graph represents acceleration due to gravity.
– Steeper line – larger gravitational acceleration
– Gentler line – smaller gravitational acceleration

Worked Example 1: Earth

S = 1/2 x (9.81) x (3)² = 1/2 x (9.81) x (9) = 44.15 m
So the rock falls $\approx$44.15 m
Final Velocity:

v = (9.81) x (3) = 29.43 m/s
Answer: After 3 s the rock has fallen $\approx$44.15 m and is moving at 29.43 m/s

Worked Example 2: Moon

The same rock is dropped from rest from a lunar platform. How far does it fall in 3 seconds on the Moon?

What is given:
– g _moon = 1.62 m/s²
– t = 3 s
– u = 0 m/s
Use the formula:

S = 1/2 x (1.62) x (3)² = 1/2 x (1.62) x (9) = 7.29 m
So the rock on the Moon falls $\approx$7.29 m in 3 s – much less than on Earth!
The rock on the Moon falls slower and covers less distance because the Moon’s gravity is weaker.

Summary:

• Gravity pulls everyhting down, but the strength of that pull varies.
• On Earth, falling objects speed up faster and fall farther in the same time.
• On the Moon, falling motion is slower because gravity is weaker.
• All free – fall motion can be predicted with equations
  

Your turn:

1. On Earth, how far does an object fall in 2 seconds (Use g = 9.8 m/s² )
2. On the Moon, how fast is an object going after 4 seconds of free fall?
   (Use g_Moon = 1.62 m/s² )
3. A feather is dropped on the Moon. How far does it fall in 5 seconds?
4. On Earth, what time does it take for a stone to fall 20 m from rest?
5. A lunar lander droops a tool from a height of 12 m above the surface.
   – How long before it hits the ground?
   – What is its final speed on the Moon?

Answers

1. Given:
   g = 9.81 m/s²
   t= 2 s
   S = 1/2 gt² =1/2 x (9.81) x (2)² = 1/2 x (9.81) x (4) = 19.62 m
   Answer : 19.62 m
2. Given:
   g_moon = 1.62 m/s²
   t= 4 s
   v = g t= (1.62) x (4) = 6.48 m/s
   Answer : 6.48 m/s
3. Given:
   g_moon = 1.62 m/s²
   t= 5 s
   S = 1/2 gt² =1/2 x (1.62) x (5)² = 1/2 x (1.62) x (25) = 20.25 m
   Answer : 20.3 m
4. Given:
   g = 9.81 m/s²
   S= 20 m
   S = 1/2 gt²
   20= 1/2 (9.81) t²
   t² = 40/9.81 = 4.08
   Answer: t$\approx$2 s
5. Given:
   g_moon = 1.62 m/s²
   S= 12 m
   Time taken to hit the ground
   S = 1/2 gt²
   12= 1/2 (1.62) t²
   t² = 24/1.62 = 14.81
   Answer: t $\approx$ 3.85
   Final Velocity:
   v = g t = (1.62) x (3.85) = 6.24 m/s
   Answer : v = 6.24 m/s