A spherical non-rotating planet (with no atmosphere) has mass m
1= 6
×10
24 kg and radius
r
1= 9000
km. A projectile of mass m
2≪
m
1 is fired from the surface of the planet at a point
A with a speed
v
A at
an angle α=30
∘ with respect to the radial direction. In its subsequent trajectory the projectile reaches a
maximum altitude at point B on the sketch. The distance from the center of the planet to the point
B is
r
2=(5/2)
r
1. Use
G=6.674×10
−11 k
g
−1
m
3
s
−2.
What is the initial speed v
A of the projectile? (in m/s)
v
A=
To find the initial speed of the projectile, we can use the principle of conservation of mechanical energy. The total mechanical energy of the projectile at point A is equal to the total mechanical energy at its maximum altitude point B.
The mechanical energy of an object can be calculated using the formula:
E = K + U,
where E is the total mechanical energy, K is the kinetic energy, and U is the gravitational potential energy.
At point A, the projectile has only kinetic energy:
K1 = 0.5 * m2 * vA^2,
where K1 is the kinetic energy at point A.
At point B, the projectile has reached its maximum altitude, so its kinetic energy is zero. The total mechanical energy is equal to the gravitational potential energy:
E = U = - G * (m1 * m2) / r2,
where U is the gravitational potential energy at point B, G is the gravitational constant, m1 is the mass of the planet, m2 is the mass of the projectile, and r2 is the distance from the center of the planet to point B.
Equating the initial and final mechanical energies:
K1 = E,
0.5 * m2 * vA^2 = - G * (m1 * m2) / r2.
Simplifying the equation:
vA^2 = - (2 * G * m1) / r2,
vA = sqrt(- (2 * G * m1) / r2).
Now we can substitute the given values to find the initial speed of the projectile:
m1 = 6 * 10^24 kg (mass of the planet),
r2 = (5/2) * r1 (distance from the center of the planet to point B),
r1 = 9000 km = 9000 * 1000 m (radius of the planet),
G = 6.674 * 10^-11 kg^-1 m^3 s^-2 (gravitational constant).
Plugging in the values:
vA = sqrt(- (2 * 6.674 * 10^-11 * 6 * 10^24) / ((5/2) * 9000 * 1000)).
Evaluating the expression:
vA ≈ sqrt(5.56 * 10^12) ≈ 2.36 * 10^6 m/s.
Therefore, the initial speed of the projectile is approximately 2.36 * 10^6 m/s.
To find the initial speed (vA) of the projectile, we can apply the conservation of energy principle.
The initial total mechanical energy of the system (planet + projectile) is equal to the final total mechanical energy when the projectile reaches its maximum altitude.
1. Initial total mechanical energy (Ei):
The initial mechanical energy is the sum of the kinetic energy and gravitational potential energy of the projectile at point A.
Ei = KEi + PEi
Since the projectile is launched from the surface of the planet, the gravitational potential energy at point A is zero. Therefore, only the kinetic energy contributes:
KEi = (1/2) * m2 * vA^2
2. Final total mechanical energy (Ef):
At the maximum altitude (point B), the kinetic energy of the projectile is zero. The only energy present is gravitational potential energy.
Ef = KEf + PEf
Since the projectile is at its maximum altitude, its kinetic energy is zero:
KEf = 0
The gravitational potential energy at point B is given by:
PEf = -GMm1 / r2
where G is the gravitational constant, M is the mass of the planet, and r2 is the distance from the center of the planet to point B.
3. Conservation of energy:
According to the conservation of energy principle, the initial and final total mechanical energies are equal:
Ei = Ef
Substitute the expressions for Ei and Ef:
(1/2) * m2 * vA^2 = -GMm1 / r2
Rearrange the equation to solve for vA:
vA^2 = (-2GMm1 / r2) * (1/m2)
Take the square root of both sides:
vA = √((-2GMm1 / r2) * (1/m2))
Substitute the given values for G, M, m1, and r2:
vA = √((-2 * 6.674 × 10^-11 kg^-1 m^3 s^-2 * 6 × 10^24 kg * (5/2) * 9000 km * 1000 m/km) / m2)
Simplify the expression:
vA = √(-(6.674 × 2 × 5 × 6 × 10) / m2) * 10
Convert the result to meters per second (m/s):
vA ≈ √(-0.200 * 10 / m2) * 10 m/s
Therefore, the initial speed vA of the projectile is approximately equal to √(-0.200 * 10 / m2) * 10 m/s.