A huge cannon is assembled on an airless planet having insignificant axial spin.The planet has a radius of 5.00 × 10⁶ m and a mass of 3.95 × 10²³ kg.The cannon fires a projectile straight up at 2000 m/s.An observation satellite orbits the planet at a height of 1000 km.What is the projectile's speed as it passes the satellite? (G = 6.67 × 10^-11 N ∙ m²/kg²) A)1500 m/s B)1380 m/s C)1610 m/s D)1280 m/s

Question

Quizplus · Accepted Answer

The projectile will reach its maximum height when its final velocity becomes zero. We can use conservation of energy to calculate this maximum height:

$Initial\,Energy = Kinetic\,Energy\,+\,Potential\,Energy$
$mv^2/2 = mgh$
where m is the mass of the projectile, v is the initial velocity, h is the maximum height.

From this equation, we can calculate the maximum height to be 20404.08 km. Since the observation satellite orbits at a height of 1000 km, we know that the projectile will pass the satellite on its way up and on its way down.

Using conservation of energy again, but now at the height of the satellite, we can calculate the velocity of the projectile as it passes the satellite on its way up:

$Total\,Energy = Kinetic\,Energy\,+\,Potential\,Energy$
$mv^2/2 - GmM_p/(r_p+h_s) = mv_s^2/2$
where Mp is the mass of the planet, rp is the radius of the planet, and hs is the height of the satellite.

Plugging in the values, we get:

$v = \sqrt{2(v_s^2 + G\,M_p/(r_p+h_s)) - v_i^2} = 1500\,m/s$

Therefore, the best choice is A) 1500 m/s.

A Huge Cannon Is Assembled on an Airless Planet Having