A 8.4 kg laundry bag is dropped from rest at an initial height of 7.00 m.
(a) What is the speed of Earth toward the bag just before the bag hits the ground? Use the value 5.98 1024 kg as the mass of Earth.
(b) Use your answer to part (a) to justify disregarding the motion of Earth when dealing with the motion of objects on Earth.
To find the speed of Earth toward the bag just before the bag hits the ground, we can use the principle of conservation of mechanical energy.
(a) The potential energy of the bag is converted into kinetic energy at the instant it hits the ground. Therefore, we can equate the potential energy at the initial height to the kinetic energy just before hitting the ground.
Potential energy (PE) = kinetic energy (KE)
The potential energy can be calculated using the formula:
PE = m * g * h
where
m = mass of the bag = 8.4 kg
g = acceleration due to gravity = 9.8 m/s^2 (approximate value on Earth)
h = initial height = 7.00 m
PE = 8.4 kg * 9.8 m/s^2 * 7.00 m
PE = 580.08 J
The kinetic energy can be calculated using the formula:
KE = 1/2 * m * v^2
where
m = mass of the bag = 8.4 kg
v = velocity of the bag just before hitting the ground (which is also the speed of Earth towards the bag)
KE = 1/2 * 8.4 kg * v^2
KE = 4.2 kg * v^2
Since the potential energy is equal to the kinetic energy:
580.08 J = 4.2 kg * v^2
Solving for v:
v^2 = 580.08 J / 4.2 kg
v^2 = 138.1143 m^2/s^2
Taking the square root of both sides:
v ≈ 11.74 m/s
The speed of Earth toward the bag just before the bag hits the ground is approximately 11.74 m/s.
(b) The speed of Earth toward the bag is relatively small compared to the average speed of objects on Earth's surface, which is approximately 1700 km/h (or about 470 m/s). Therefore, when dealing with the motion of objects on Earth, we can disregard the motion of Earth as its speed is much larger in comparison.
To determine the speed of Earth toward the laundry bag just before it hits the ground, we can use the principle of conservation of energy.
(a) First, we need to find the potential energy (PE) of the laundry bag at the initial height. The formula for potential energy is given by:
PE = mgh
where m is the mass of the bag, g is the acceleration due to gravity, and h is the initial height.
Given:
m = 8.4 kg
g = 9.8 m/s^2 (approximately)
h = 7.00 m
Plugging in these values, we get:
PE = (8.4 kg) * (9.8 m/s^2) * (7.00 m)
= 570.24 Joules
Now, according to the principle of conservation of energy, the total mechanical energy (TE) of the system remains constant. It is the sum of the potential energy (PE) and the kinetic energy (KE) of the laundry bag just before it hits the ground.
TE = PE + KE
But at the ground, the height (h) is zero (since the bag has hit the ground). So the potential energy becomes zero as well.
TE = KE
Therefore, we can equate the total mechanical energy (570.24 Joules) to the kinetic energy of the laundry bag:
TE = 0.5 * mv^2
where v is the velocity of the bag just before it hits the ground.
Rearranging the equation to solve for v:
v = sqrt(2 * TE / m)
Plugging in the values, we get:
v = sqrt(2 * 570.24 J / 8.4 kg)
≈ 10.25 m/s
So, the speed of Earth toward the bag just before the bag hits the ground is approximately 10.25 m/s.
(b) To justify disregarding the motion of Earth when dealing with the motion of objects on Earth, we can consider the massive difference in mass between the Earth and the laundry bag.
The mass of Earth (5.98 × 10^24 kg) is significantly larger than the mass of the laundry bag (8.4 kg). This huge difference in mass means that the motion of Earth has negligible effect on the motion of objects on its surface, including the laundry bag. The acceleration due to Earth's motion is so small compared to gravitational acceleration that it can be disregarded in most cases. Therefore, we can safely ignore the motion of Earth when considering the motion of objects on Earth.
mgh =mv²/2.
v=sqrt (2•g•h) = sqrt(2•9.8•7) = 11.71 m/s
The speed of the bag respectively the Earth is equal to the speed of Earth respectively the bag.