Take the mass of the Earth to be 5.98 ×
1024 kg.
If the Earth’s gravitational force causes a
falling 66 kg student to accelerate downward
at 9.8 m/s2, determine the upward accelera-
tion of the Earth during the student’s fall.
Well, the Earth may be heavy, but it doesn't have to worry about gaining weight! So, let's calculate the upward acceleration of our Earth while our unlucky student is falling.
We know the mass of the Earth (5.98 × 10^24 kg) and the mass of the student (66 kg). The force of gravity acting on the student is given by F = m*a, where m is the mass and a is the acceleration.
So, in this case, the force acting on the student is F = 66 kg * 9.8 m/s^2 (downward direction).
Now, according to Newton's third law of motion, the Earth also exerts an equal and opposite force on the student. So, the Earth's force acting on the student is -F, which is -66 kg * 9.8 m/s^2 (upward direction).
To determine the upward acceleration of the Earth, we can use the formula F = m*a, rearranged to a = F/m. Plugging in the values, we get a = (-66 kg * 9.8 m/s^2) / (5.98 × 10^24 kg).
But wait a minute, this value is going to be incredibly small because the mass of the Earth is so large compared to the student! So small, in fact, that it's practically negligible! In other words, the Earth's upward acceleration during the student's fall is pretty much zilch. The Earth is too massive to stretch a muscle, even with falling students!
To determine the upward acceleration of the Earth during the student’s fall, we can use Newton's third law of motion, which states that every action has an equal and opposite reaction.
Step 1: Calculate the force experienced by the student.
The force experienced by the student can be calculated using Newton's second law of motion, which states that force equals mass times acceleration (F = m*a).
Given:
Mass of the student (m) = 66 kg
Acceleration of the student (a) = 9.8 m/s^2
F = m*a
F = 66 kg * 9.8 m/s^2
F = 646.8 N
Step 2: Calculate the force experienced by the Earth.
According to Newton's third law, the force experienced by the Earth is equal in magnitude but opposite in direction to the force experienced by the student.
F (Earth) = -F (student)
F (Earth) = -646.8 N
Step 3: Calculate the acceleration of the Earth.
The acceleration of the Earth can be determined using the formula for gravitational force, which states that force equals the gravitational constant (G) times the product of the masses divided by the square of the distance between the two masses (F = G*(m1*m2)/r^2).
Given:
Mass of the Earth (m1) = 5.98 x 10^24 kg
Mass of the student (m2) = 66 kg
Gravitational constant (G) = 6.67430 x 10^-11 N(m/kg)^2
F (Earth) = G*(m1*m2)/r^2
-646.8 N = (6.67430 x 10^-11 N(m/kg)^2) * (5.98 x 10^24 kg * 66 kg)/r^2
Simplifying the equation, we can solve for the acceleration of the Earth (a (Earth)):
a (Earth) = (-646.8 N * r^2) / (6.67430 x 10^-11 N(m/kg)^2 * 5.98 x 10^24 kg * 66 kg)
Please note that the value of r (distance between the student and the center of the Earth) is not provided in the question. Without this information, it is not possible to determine the exact magnitude of the upward acceleration of the Earth during the student’s fall.
To determine the upward acceleration of the Earth during the student's fall, we can use Newton's third law of motion, which states that for every action, there is an equal and opposite reaction.
Here's how you can calculate the upward acceleration of the Earth:
1. Recall Newton's third law: The force exerted on the student due to the Earth's gravity is equal in magnitude and opposite in direction to the force exerted on the Earth by the student.
F_student = F_earth
2. Calculate the force exerted on the student:
F_student = m_student * g
where m_student is the mass of the student (66 kg) and g is the acceleration due to gravity (9.8 m/s^2).
F_student = 66 kg * 9.8 m/s^2 = 646.8 N
3. Since the force exerted by the student on the Earth is equal in magnitude but opposite in direction, the force exerted by the Earth on the student is also 646.8 N.
4. Apply Newton's third law:
F_student = F_earth => 646.8 N = m_earth * a_earth
where m_earth is the mass of the Earth (5.98 × 10^24 kg) and a_earth is the upward acceleration of the Earth during the student's fall.
5. Rearrange the equation to solve for a_earth:
a_earth = F_student / m_earth
a_earth = 646.8 N / 5.98 × 10^24 kg
6. Calculate the upward acceleration of the Earth:
a_earth ≈ 1.083 × 10^(-22) m/s^2 (approximately)
Therefore, the upward acceleration of the Earth during the student's fall is approximately 1.083 × 10^(-22) m/s^2.