A mass of 20 kg is held 1.20 107 metres above the surface of a planet that has a radius of 1.50 107 metres and a mass of 4.50 1028 kg.
a) If the mass were to be dropped freely to hit the surface of the planet, with what velocity would it collide with the planet?
Please help...
To calculate the velocity with which the mass will collide with the planet, we can use the principle of conservation of mechanical energy.
The potential energy of the mass when it is at a height h above the surface of the planet is given by:
PE = mgh
Where:
m = mass of the object = 20 kg
g = acceleration due to gravity on the planet's surface. Since the planet's mass and radius are given, we can calculate g using the formula:
g = (G * M) / (R^2)
Where:
G = gravitational constant = 6.67430 × 10^-11 N(m/kg)^2
M = mass of the planet = 4.50 × 10^28 kg
R = radius of the planet = 1.50 × 10^7 m
Now, we can find the potential energy at the given height:
PE = (20 kg) * (9.8 m/s^2) * (1.20 × 10^7 m)
Next, we can use the principle of conservation of mechanical energy to find the velocity with which the mass will collide with the planet. At the surface of the planet, all the potential energy will have been converted to kinetic energy. Therefore, the kinetic energy of the mass when it hits the surface is given by:
KE = 0.5 * m * v^2
where v is the velocity of the mass.
Since mechanical energy is conserved, we can equate the potential energy to the kinetic energy:
PE = KE
Substituting the respective formulas, we get:
(20 kg) * (9.8 m/s^2) * (1.20 × 10^7 m) = 0.5 * (20 kg) * v^2
We can now solve for v by rearranging the equation and isolating v:
v^2 = (2 * (20 kg) * (9.8 m/s^2) * (1.20 × 10^7 m)) / (20 kg)
v^2 = (2 * (9.8 m/s^2) * (1.20 × 10^7 m))
Taking the square root of both sides to solve for v, we get:
v = √[(2 * (9.8 m/s^2) * (1.20 × 10^7 m))]
Now, we can substitute the given values and solve for v:
v = √[(2 * (9.8 m/s^2) * (1.20 × 10^7 m))]
v = √[23.52 × 10^7 m^2/s^2]
v = √[23.52] m/s
v ≈ 4.85 × 10^3 m/s
Therefore, the mass would collide with the planet with a velocity of approximately 4.85 × 10^3 m/s.