We release an object of mass M at zero speed at a distance of 2*10^6 Km from the center of the earth. What is the speed at which the object hits the earth?
G=6.67*10^-11; g= 9.8 m/s; M=5.97*10^24;
r=6.3781*10^6
Suppose the mass of the object is M= 500 kg. What is the kinetic energy of this object when it hits the earth?
Thanks for your help.
M= 5.97•10²⁴ kg
m=500 kg
r = 6.378•10⁶ m.
R= 2• 0⁶ km= 2• 10⁹ m
GmM/r-GmM/R = mv²/2
GM(1/r -1/R) = v²/2
v=sqrt{2GM(r-R)/rR}=
=sqrt {2•6.67•10⁻¹¹• 5.97•10²⁴(2• 10⁹ - 6.3781•10⁶)/2• 10⁹•6.3781•10⁶}=
=1.1•10⁴ m/s
mv²/2 = 500•(1.1•10⁴)²/23.025 •10¹º J
Thanks Elena for your help. I have been having trouble with some of these equations.
To find the speed at which the object hits the earth, we can use the concept of gravitational potential energy.
1. First, let's calculate the potential energy of the object at the initial position using the formula:
Potential Energy = (G * M * m) / r,
where G is the gravitational constant (6.67 * 10^-11), M is the mass of the Earth (5.97 * 10^24), m is the mass of the object, and r is the initial distance from the center of the Earth (2 * 10^6 km converted to meters, i.e., 2 * 10^9 m).
Potential Energy = (6.67 * 10^-11 * 5.97 * 10^24 * 500) / (2 * 10^9 + 6.3781 * 10^6).
2. Next, let's calculate the kinetic energy of the object when it hits the Earth. Since it starts from rest, all the potential energy will convert into kinetic energy.
Kinetic Energy = Potential Energy.
3. Finally, we can use the kinetic energy formula to find the speed (v) at which the object hits the Earth:
Kinetic Energy = 0.5 * m * v^2,
where m is the mass of the object.
Now, let's calculate the answers to your questions.
1. Speed at which the object hits the Earth:
- Calculate the potential energy:
Potential Energy = (6.67 * 10^-11 * 5.97 * 10^24 * 500) / (2 * 10^9 + 6.3781 * 10^6).
- Calculate the kinetic energy:
Kinetic Energy = Potential Energy.
- Calculate the speed:
Speed = sqrt((2 * Kinetic Energy) / m).
2. Kinetic energy of the object when it hits the Earth:
- The value of the kinetic energy is equal to the potential energy calculated above.
By following these steps, you will be able to find the speed at which the object hits the Earth and the kinetic energy of the object.