Two hypothetical planets of masses m1 and m2 and radii r1 and r2, respectively, are nearly at rest when they are an infinite distance apart. Because of their gravitational attraction, they head toward each other on a collision course. Note: Both the energy and momentum of the isolated two planet system are constant.
(b) Find the kinetic energy of each planet just before they collide, taking m1 = 1.80* 10^24 kg, m2 = 9.00* 10^24 kg, r1 = 3.40* 10^6 m, and r2 = 5.20* 10^6 m.
please could you help me get kinetic energy
k1 = ?
k2 = ?
please thank you
Since the masses are in a 1:5 ratio, and the total momentum remains zero, the large mass will have 1/5 the velocity of the smaller mass at all times.
They collide when the separation distance is d = r1 + r2 = 5.6*10^6 m
At that time, the total kinetic energy equals the potential energy loss, which is
KE = G*M1*M2/d
With 1/5 of the mass of M2, M1 will have 5 times the velocity and 5 times its kinetic energy, giving it 5/6 of the total KE. M2 will have the other 1/6.
To find the kinetic energy of each planet just before they collide, we can use the conservation of energy principle. The initial potential energy of the two planets when they are at an infinite distance apart will be equal to the sum of their final kinetic energies before they collide.
The potential energy between two objects is given by the equation:
U = -G * (m1 * m2) / r
Where:
U is the potential energy
G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
m1 and m2 are the masses of the planets
r is the distance between the centers of the two spheres
Now, let's calculate the potential energy between the two planets:
U = -G * (m1 * m2) / r
U = - (6.67430 × 10^-11 m^3 kg^-1 s^-2) * ((1.80 × 10^24 kg) * (9.00 × 10^24 kg)) / (3.40 × 10^6 m + 5.20 × 10^6 m)
U ≈ -4.82 × 10^33 J
Since the sum of the initial potential energy and the final kinetic energy is constant, the kinetic energy just before they collide will be equal to the absolute value of the potential energy:
k1 = |U|
k2 = |U|
Now let's calculate the kinetic energy for each planet:
k1 = |U| = 4.82 × 10^33 J
k2 = |U| = 4.82 × 10^33 J
Therefore, the kinetic energy of each planet just before they collide is approximately 4.82 × 10^33 J.
To find the kinetic energy of each planet just before they collide, we need to use the conservation of mechanical energy. By definition, the mechanical energy of a system remains constant as long as no external forces act upon it.
The total mechanical energy of a two-planet system consists of the sum of kinetic energy and gravitational potential energy. Thus, we can write the equation as follows:
Total mechanical energy before collision = Total mechanical energy after collision
Since the planets are initially at rest, their initial kinetic energy is zero. Therefore, the equation becomes:
Gravitational potential energy before collision = Final kinetic energy after collision
The gravitational potential energy between two objects is given by the equation:
Potential energy = -G * m1 * m2 / r
Where G is the gravitational constant (approximately 6.674 × 10^(-11) N m^2 / kg^2), m1 and m2 are the masses of the two planets, and r is the distance between them.
Let's calculate the potential energy before the collision:
Potential energy before collision = - (G * m1 * m2) / r
Now, since the initial kinetic energy of both planets is zero, the final kinetic energy after the collision will be equal to the total mechanical energy before the collision. Hence, we can rewrite the equation as:
Final kinetic energy after collision = - (G * m1 * m2) / r
Now, let's substitute the given values into the equation:
m1 = 1.80 × 10^24 kg
m2 = 9.00 × 10^24 kg
r = distance between the planets = r1 + r2 = 3.40 × 10^6 m + 5.20 × 10^6 m
G = 6.674 × 10^(-11) N m^2 / kg^2
Final kinetic energy after collision = - (G * m1 * m2) / r
Now plug in the values and solve for the final kinetic energy for each planet.