A spherical non-rotating planet (with no atmosphere) has mass m1= 3 ×1024 kg and radius r1= 8000 km. A projectile of mass m2≪m1 is fired from the surface of the planet at a point A with a speed vA 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 r2=(5/2)r1. Use G=6.674×10−11 kg−1m3s−2.
What is the initial speed vA of the projectile? (in m/s)
vA=
3220
Its incorrect.. Can you plz give how to find it?
To find the initial speed vA of the projectile, we can start by applying the conservation of mechanical energy.
At point A, the projectile has both kinetic energy and gravitational potential energy. At the maximum altitude point B, the projectile has only gravitational potential energy.
The conservation of mechanical energy equation is given by:
KE_A + PE_A = KE_B + PE_B
Since the projectile is fired from the surface of the planet, the gravitational potential energy at point A is zero (PE_A = 0).
At point B, the gravitational potential energy is given by:
PE_B = -G * (m1 * m2) / r2
where G is the universal 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.
The kinetic energy at point B can be calculated as:
KE_B = (1/2) * m2 * vB^2
where vB is the velocity of the projectile at point B.
Since the projectile reaches its maximum altitude at point B, the velocity vB is zero (vB = 0) because the projectile momentarily stops moving vertically.
Substituting the values into the conservation of energy equation, we have:
KE_A + 0 = (1/2) * m2 * 0^2 - G * (m1 * m2) / r2
Simplifying the equation, we get:
KE_A = G * (m1 * m2) / r2
Since the kinetic energy at point A is given by:
KE_A = (1/2) * m2 * vA^2
where vA is the initial speed of the projectile, we can equate both equations:
(1/2) * m2 * vA^2 = G * (m1 * m2) / r2
Simplifying further, we can solve for vA:
vA^2 = 2 * (G * m1 / r2)
Taking the square root of both sides, we get:
vA = sqrt(2 * (G * m1 / r2))
Now, substituting the given values, we have:
vA = sqrt(2 * (6.674×10^-11 kg^-1m^3s^-2 * 3 ×10^24 kg) / (5/2 * 8000 km))
Note: We need to convert the radius from kilometers to meters in order to get the correct units for velocity.
Converting 8000 km to meters, we have:
8000 km = 8000 * 1000 m = 8 * 10^6 m
Substituting the values into the equation, we can calculate the initial speed vA of the projectile.