A spherical non-rotating planet (with no atmosphere) has mass m1= 5 ×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.

Question

A spherical non-rotating planet (with no atmosphere) has mass m1= 5 ×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)

Answer 1

http://web.mit.edu/8.01t/www/materials/InClass/IC_Sol_W13D1-8.pdf

Answer 2

To find the initial speed of the projectile (vA), we need to use the principle of conservation of mechanical energy. The mechanical energy of the system (projectile + planet) is conserved in the absence of external forces like air resistance.

The initial mechanical energy of the system is given by the sum of the kinetic energy and the gravitational potential energy at point A. The final mechanical energy at point B is the sum of the kinetic energy and the gravitational potential energy at point B.

Since the projectile is fired vertically with an angle of 30 degrees from the radial direction, we can assume the initial kinetic energy in the radial direction is negligible compared to the initial kinetic energy in the vertical direction.

The gravitational potential energy can be calculated using the formula:

Gravitational Potential Energy = - G * (mass1 * mass2) / distance.

At point A:
Initial kinetic energy = 0.5 * mass2 * vA^2
Gravitational potential energy = - G * (mass1 * mass2) / r1

At point B:
Final kinetic energy = 0.5 * mass2 * vB^2
Gravitational potential energy = - G * (mass1 * mass2) / r2

Since the initial and final kinetic energy of the projectile cancel each other out, we only consider the gravitational potential energy for this calculation.

Setting the initial and final gravitational potential energies equal to each other:

- G * (mass1 * mass2) / r1 = - G * (mass1 * mass2) / r2

Simplifying the equation:

r2 / r1 = 5/2

Solving for r2:

r2 = 2.5 * r1

Now we can substitute the values into the equation:

(2.5 * r1) / r1 = 5/2

2.5 = 5/2

Since the equation is true, the given value for r2 satisfies this condition. Therefore, the given information is consistent.

So, we have the following equation:

- G * (mass1 * mass2) / r1 = - G * (mass1 * mass2) / (5/2 * r1)

Canceling out the common factors:

1 / r1 = 1 / (5/2 * r1)

Multiplying both sides by r1:

1 = 2/5

The equation is not consistent, which means there is an error in the given information or the calculation. Please recheck the values and equations provided.