You stand on the seat of a chair and then hop off.

During the time you are in flight down to the floor, the Earth is lurching up toward you with an acceleration a. If your mass is 52 kg, what is the value of a? Visualize the Earth as a perfectly solid object.

why isnt acceleration = 9.81m/s^2 due to gravitational force?

That IS the acceleration due to gravity of the person, near the surface of the earth, if the person is free to fall.

It is a bit of a trick question. Actually, the person and the earth approach each other at different acceleration rates. They are reacting to equal and opposite force between them, but the earth "lurches upward" at a much lower rate,
a = g (m/M)
where M is the mass of the earth and m is the mass of the person.

a = 9.81 m/s^2*(52/5.97*10^24)
= 8.5*10^-28 m/s^2

To find the value of acceleration (a), we need to apply Newton's second law of motion. Newton's second law states that the force (F) acting on an object is equal to the mass (m) of the object multiplied by its acceleration (a).

In this scenario, when you hop off the chair and are in flight, the only force acting on you is the force due to gravity. The force due to gravity is given by the formula F = mg, where m is the mass of the object and g is the acceleration due to gravity (approximately 9.8 m/s² on Earth).

However, in this case, the Earth is lurching up towards you with an acceleration a. So, the force acting on you is not just due to gravity but also due to the Earth's acceleration.

Since the question mentions that the Earth is a perfectly solid object, we can consider the Earth's mass M and its acceleration towards you as the force acting on you. Therefore, the force equals F = Ma.

Now, relate the two forces together. The force due to gravity acting on you (mg) is equal to the force due to the acceleration of the Earth (Ma). Therefore, we can write the equation as:
mg = Ma

Given that your mass (m) is 52 kg, we can rearrange the equation to solve for a:
a = (mg) / M

Substituting the known values, where g is approximately 9.8 m/s² and the mass of the Earth M is approximately 5.972 × 10^24 kg, we have:
a = (52 kg × 9.8 m/s²) / (5.972 × 10^24 kg)

Now, calculate the value of a using a calculator:
a ≈ 8.47 × 10^-23 m/s²

Therefore, the value of acceleration (a) in this scenario is approximately 8.47 × 10^-23 m/s².