HSC Physics Newton's Cannon

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A tutorial sheet of harder problems on Newton's cannon is given below. Neglect air resistance, the rotation of the Earth and the movement of the Earth about the Sun.

  1. Define escape speed.
  2. Derive the equation for escape speed in terms of G, M and r.
  3. Express escape speed in terms of g, G and r.
  4. Does escape speed depend on the mass of the projectile m? Explain why in words.
  5. Does escape speed depend on the launch angle?
  6. Calculate the value of the escape speed from the surface of the Earth.
  7. A cannon on a tall mountain fires a projectile horizontally at a small speed. Explain why the projectile falls to the Earth.
  8. A projectile is fired from the surface of the Earth at 5 kms-1 at 45° to the horizontal. Does this projectile go into orbit around the Earth?
  9. The escape speed from the surface of a planet of radius R is V. A projectile is thrown vertically upwards at a speed V/2. Determine the maximum height reached by the projectile.
  10. A tall mountain has an altitude h. The radius of the Earth is R and its mass is M. A cannon fires a shell horizontally at a speed u from the top of the mountain. Find the speed with which the shell strikes the Earth.
  11. *A projectile is given an initial speed of 5 km/s at 45° to the horizontal from the surface of the Earth. Find the range on the surface of the Earth and the time of flight. What launch angle gives maximum range on the surface of the Earth? Take g=9.81 m/s2 and R=6378 km [3121km, 1052s, 38°, 3219km 2Rsin-1[v2/(2gR-v2)]]
  12. *In the previous question the projectile is fired from a point on the equator towards the east. Taking into account the rotation of the Earth, how are the previous answers modified? [decreases (2706km), takes longer (1135s), lower angle (33.6°), decreases (2972km)]