A spherical non-rotating planet (with no atmosphere) has mass 4 kg and radius 9000 km. A projectile of mass is fired from the surface of the planet at a point with a speed at an angle with respect to the radial direction. In its subsequent trajectory the projectile reaches a maximum altitude at point on the sketch. The distance from the center of the planet to the point is . Use .
What is the initial speed of the projectile? (in m/s)
To calculate the initial speed of the projectile, we need to use the gravitational potential energy and kinetic energy at the surface of the planet.
The gravitational potential energy (PE) is given by:
PE = -G * (m1 * m2) / r
where G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2), m1 and m2 are the masses of the planet and the projectile respectively, and r is the distance between the planet's center and the point where the projectile is fired.
The kinetic energy (KE) is given by:
KE = 0.5 * m2 * v^2
where v is the initial speed of the projectile.
Since the planet is non-rotating and has no atmosphere, the total mechanical energy (E) of the projectile remains constant throughout its trajectory:
E = PE + KE
At the maximum altitude point, the kinetic energy is zero, and therefore the total mechanical energy is equal to the gravitational potential energy:
E = PE = -G * (m1 * m2) / r
Substituting the given values, we have:
E = -G * (m1 * m2) / r
E = 0.5 * m2 * v^2
Setting these two equations equal to each other, we can solve for v:
-G * (m1 * m2) / r = 0.5 * m2 * v^2
Simplifying, we get:
v^2 = -2 * G * m1 / r
Taking the square root of both sides, we get:
v = √(-2 * G * m1 / r)
Now we can substitute the given values and calculate the initial speed:
v = √(-2 * (6.67430 × 10^-11 m^3 kg^-1 s^-2) * (4 kg) / (9000 km = 9 × 10^6 m))
v = √(-1.49967 × 10^-18 m^2 kg^-1 s^-2)
v ≈ 1.22458 × 10^-9 m/s
Therefore, the initial speed of the projectile is approximately 1.22458 × 10^-9 m/s.
To find the initial speed (v0) of the projectile, we need to consider the gravitational potential energy and kinetic energy of the system.
1. First, we need to calculate the gravitational potential energy (PE) at the surface of the planet. The formula for gravitational potential energy is given by:
PE = (-GMm) / R
where:
G = universal gravitational constant (6.674 × 10^-11 m^3 kg^-1 s^-2)
M = mass of the planet (4 kg)
m = mass of the projectile (unknown)
R = radius of the planet (9000 km or 9,000,000 m)
Since the planet is non-rotating and has no atmosphere, there is no change in gravitational potential energy as the projectile moves away from the surface.
2. Next, we consider the kinetic energy (KE) at the highest point in the trajectory. At the highest point, the projectile will momentarily come to rest before falling back down. Therefore, the kinetic energy is zero at this point.
KE = 0
3. The total mechanical energy (E) of the system is the sum of the potential and kinetic energies:
E = PE + KE
Since the total mechanical energy is conserved, we can equate the total energy at the surface of the planet to the total energy at the highest point:
PE = KE
-GMm / R = 0
This equation implies that the projectile's mass (m) cancels out, so it is not necessary for the calculation of initial speed (v0).
4. Now we can solve for the initial speed (v0) using the equation for kinetic energy:
KE = (1/2) mv0^2
Since the kinetic energy is zero at the highest point:
0 = (1/2) mv0^2
Solving for v0:
v0^2 = 0 / (1/2) = 0
v0 = √0 = 0 m/s
Therefore, the initial speed (v0) of the projectile is 0 m/s.