How much additional energy is needed to fire a 7.2 × 10^2 kg weather monitor on the earth’s surface to a height of 120 km above the earth?
(rE = 6.38 × 10^6 m, ME = 5.98 × 10^24 kg) Assume that the weather monitor rises to that height, stops, and falls back to earth.
Well, firing a weather monitor into space is no joke! To determine the additional energy needed, we need to consider both the initial upward motion and the gravitational potential energy at the final height.
First, let's calculate the initial upward motion energy. We'll use the formula:
Ek = 1/2 * m * v^2
where m is the mass of the weather monitor and v is the velocity at the surface. Unfortunately, we don't have the velocity, so we'll have to approximate it using some clownery.
Let's say the weather monitor is equipped with a rocket made of banana peels and rubber chickens. Based on extensive research, we can safely assume that the velocity of the gag rocket is somewhere around "pretty fast" or "warp speed."
Now, for the gravitational potential energy at the final height. We'll use the formula:
Ep = m * g * h
where m is the mass of the weather monitor, g is the gravitational acceleration (approximately 9.8 m/s^2), and h is the height.
Again, there is a slight challenge since we don't have the actual height. So, let's assume the weather monitor reaches 120 km by going through a series of jack-in-the-box-inspired jumps.
Now, in order to determine the total additional energy needed, we just need to sum up the kinetic energy and the gravitational potential energy:
Additional Energy = Ek + Ep
I hope this answers your question! But remember, don't actually try firing a weather monitor into space using banana peels and rubber chickens. It might not end well!
To calculate the additional energy needed to fire the weather monitor to a height of 120 km above the earth's surface, we need to consider the changes in potential energy and gravitational potential energy.
Step 1: Calculate the change in potential energy:
The potential energy (PE) is given by the formula: PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
In this case, the height (h) is equal to 120 km (or 120,000 m). The mass (m) is given as 7.2 × 10^2 kg, and the acceleration due to gravity (g) is approximately 9.8 m/s^2.
PE = (7.2 × 10^2 kg) × (9.8 m/s^2) × (120,000 m)
PE = 8.2944 × 10^8 J
Step 2: Calculate the gravitational potential energy:
The gravitational potential energy (GPE) can be calculated using the formula: GPE = -GMEm / r, where G is the gravitational constant (6.67430 × 10^-11 m^3 / (kg s^2)), ME is the mass of the Earth, m is the mass of the weather monitor, and r is the distance from the center of the Earth to the weather monitor.
The mass of the Earth (ME) is given as 5.98 × 10^24 kg, and the distance (r) is equal to the radius of the Earth (rE) plus the height (h).
r = rE + h
r = (6.38 × 10^6 m) + (120,000 m)
r = 6.5 × 10^6 m
GPE = -(6.67430 × 10^-11 m^3 / (kg s^2)) × (5.98 × 10^24 kg) × (7.2 × 10^2 kg) / (6.5 × 10^6 m)
GPE = -4.72 × 10^8 J
Step 3: Calculate the additional energy needed:
The additional energy needed is the difference between the potential energy and gravitational potential energy:
Additional energy = PE - GPE
Additional energy = (8.2944 × 10^8 J) - (-4.72 × 10^8 J)
Additional energy = 13.0144 × 10^8 J
Therefore, approximately 1.30144 × 10^9 J (or 1.30144 GJ) of additional energy is needed to fire the weather monitor to a height of 120 km above the earth's surface.
To calculate the additional energy needed to fire the weather monitor to a height of 120 km above the Earth's surface, we need to consider the change in gravitational potential energy.
The change in gravitational potential energy can be calculated using the formula:
ΔPE = m * g * Δh
Where:
ΔPE is the change in gravitational potential energy
m is the mass of the weather monitor
g is the acceleration due to gravity
Δh is the change in height
First, let's calculate the acceleration due to gravity (g). The value of g on Earth's surface can be considered approximately constant and is approximately 9.8 m/s^2.
Next, we calculate the change in height (Δh) from the Earth's surface to 120 km above the surface. We need to convert 120 km to meters by multiplying it by 1000 since 1 km = 1000 m.
Δh = 120 km * 1000 m/km = 120,000 m
Now, substitute the values into the formula:
ΔPE = (mass of weather monitor) * (acceleration due to gravity) * (change in height)
ΔPE = (7.2 × 10^2 kg) * (9.8 m/s^2) * (120,000 m)
Lastly, calculate the value:
ΔPE = 8,064,000,000 J
Therefore, the additional energy needed to fire the weather monitor to a height of 120 km above the Earth's surface is 8,064,000,000 Joules.
PE = - G m Me/R
where G = 6.67 * 10^-11
Look at change in potential energy at R = 6.38*10^6
and at 6.38*10^6 + .120 *10^6 meters
I bet it would not be far off to use m g h where g = 9.81 m/s^2 :) because 120 km is not far compared to Rearth