An astronaut on a strange planet finds that she can jump a maximum horizontal distance of 10.0 m if her initial speed is 2.60 m/s. What is the free-fall acceleration on the planet?

BOBO KA

To determine the free-fall acceleration on the planet, we can use the concept of projectile motion. When the astronaut jumps, she follows a curved path known as a projectile. The horizontal and vertical motions of the astronaut are independent of each other.

Given:
Max horizontal distance (range) = 10.0 m
Initial speed (vertical velocity) = 2.60 m/s

We know that the horizontal motion is not affected by gravity, and hence, the horizontal component of velocity remains constant throughout the projectile motion.

Using the formula for horizontal distance (range) in projectile motion:
Range = Horizontal velocity × Time of flight

Since the horizontal velocity remains constant, we can substitute:
Range = Horizontal velocity × Time of flight
10.0 m = Horizontal velocity × Time of flight

To find the time of flight, we need to determine the total time the astronaut spends in the air. During free fall, the only force acting on the astronaut is the gravitational force. Thus, we can use the equation for vertical displacement in free fall to calculate the time of flight.

The formula for vertical displacement in free fall is given by:
Vertical displacement = (1/2) × acceleration due to gravity × Time²

In this case, the vertical displacement can be equated to zero since the astronaut lands at the same height as when she took off.

Substituting the known values into the equation:
0 = (1/2) × acceleration due to gravity × Time of flight²

Simplifying the equation:
0 = (1/2) × acceleration due to gravity × (2 × Time of flight)

Since Time of flight is common to both sides of the equation, we can simplify it further:
0 = acceleration due to gravity × (Time of flight)

Thus, we have:
0 = acceleration due to gravity

From this equation, we can conclude that the acceleration due to gravity on the strange planet is zero. This suggests that either the planet has no mass and hence no gravitational force, or it is experiencing a unique phenomenon not encountered in our usual understanding of physics.