91Ó°ÊÓ

When discussing projectile motion, one of the key factors involved is the acceleration due to gravity. On Earth, this is approximately constant at \(9.81 \, \text{m/s}^2\). This value represents how quickly an object will accelerate downwards when in free fall, regardless of its horizontal movement.
The acceleration due to gravity (\(g\)) is crucial in determining how long an object takes to hit the ground from a height. In the original exercise, the differing accelerations on Earth compared to Planet \(X\) affected how far the ball traveled horizontally given the same initial conditions.
This concept explains why, if gravity is weaker on Planet \(X\), as calculated to be approximately \(1.287 \, \text{m/s}^2\), the ball travels a greater horizontal distance, because it falls more slowly, staying in motion for longer before hitting the ground.

Free fall equations are fundamental in calculating how long an object in projectile motion takes to fall from a certain height. They help relate the vertical motion to time by considering gravity's effect on the object.
The main equation is \( h = \frac{1}{2} g t^2 \), where \( h \) is the height from which the object is dropped, \( g \) is the gravitational acceleration, and \( t \) is the time taken to fall.

• This equation assumes the initial vertical speed is zero when an object falls from rest.
• The equation helps determine the time \( t \), which is crucial for calculating how far an object can travel horizontally before hitting the ground.

In the exercise, the height \( h \) was unknown, but by using the free fall equations, we could relate \( h \) to time and eventually solve for the horizontal distance traveled under different gravitational forces.

In projectile motion, horizontal motion is considered separately from vertical motion. Although the object is affected by gravity, its horizontal speed remains constant (as long as air resistance is ignored).
For horizontal motion, the formula \( D = v_0 \times t \) describes how distance \( D \) is proportional to the initial horizontal speed \( v_0 \) and the time \( t \) it remains in motion. This equation can be crucial in predicting where an object will land.

• At the same initial speed, the time period for which the object stays airborne determines how far it travels horizontally.
• Hence, if the time of flight increases due to lower gravitational acceleration, the horizontal distance traveled increases as well.

In the given exercise, these principles explain why the same ball rolled a further distance on Planet \(X\) than on Earth, owing to the difference in gravity, thereby increasing its in-air time.