A rocket is launched straight up from the earth's surface at a speed of 1.80×104 m/s^2.
a)What is its speed when it is very far away from the earth?
Energy dictates that some of the KE will go to increasing gravitational PE.
initial KE=finalKE+(GMe/re-GMe/infinity)
initialKE=finalKe+GMe/re
solve for final Ke, then final velocity
is Me the mass of the earth?
and re the radius of the earth?
To find the speed of the rocket when it is very far away from the earth, we need to consider the concept of escape velocity.
Escape velocity is the minimum velocity an object must reach in order to escape the gravitational pull of a planet or celestial body. For Earth, the escape velocity is approximately 11.2 km/s or 1.12 × 10^4 m/s.
Since the rocket's initial speed is given as 1.80 × 10^4 m/s, which is greater than the escape velocity, it means that the rocket has enough speed to escape Earth's gravitational pull. Therefore, when the rocket is very far away from the Earth, its speed will remain constant at 1.80 × 10^4 m/s.
To determine the speed of the rocket when it is very far away from the Earth, we need to consider the principle of conservation of energy.
When the rocket is far away from the Earth, we can assume that the gravitational potential energy it had initially on the Earth's surface has been converted into kinetic energy. The total mechanical energy (E) of the rocket is given by the sum of its kinetic energy (K) and gravitational potential energy (U):
E = K + U
Since the rocket is launched vertically (upwards), we can neglect the effects of air resistance, and the only significant force acting on the rocket is gravity. The work done by gravity as the rocket moves from the Earth's surface to its farthest point is equal to the change in gravitational potential energy:
ΔU = -W
Where ΔU is the change in gravitational potential energy and W is the work done by gravity.
The work done by gravity can be calculated using the formula:
W = mgh
Where m is the mass of the rocket, g is the acceleration due to gravity (approximately 9.8 m/s^2 near the Earth's surface), and h is the height (distance) from the Earth's surface to the rocket when it is very far away.
As the rocket moves from the Earth's surface to very far away, the height h becomes infinitely large, and the gravitational potential energy U approaches zero. Therefore, the change in gravitational potential energy ΔU is approximately equal to zero.
ΔU = 0
Since the change in gravitational potential energy is zero, the work done by gravity W is also zero.
W = 0
Using the formulas above, we have:
ΔU = -W
0 = -W
This implies that the total mechanical energy of the rocket remains constant (E = constant). Since the total mechanical energy is the sum of kinetic and gravitational potential energies, we can write:
E = K + U
Since the change in gravitational potential energy is zero, U = 0.
E = K
Therefore, the total mechanical energy E is equal to the kinetic energy K.
Now, we can calculate the speed of the rocket when it is very far away. The kinetic energy (K) is given by the formula:
K = (1/2)mv^2
Where m is the mass of the rocket and v is its velocity.
Since we know the velocity of the rocket when it was launched from the Earth's surface (1.80 x 10^4 m/s), we can substitute the given values into the formula for kinetic energy:
K = (1/2)m(1.80 x 10^4 m/s)^2
To determine the speed of the rocket when it is very far away, we need to know the mass of the rocket or have additional information about it. Without that information, it is not possible to calculate the speed of the rocket when it is very far away from the Earth.