A particle is projected from the origin with speed u in a direction making an angle α with the horizontal. The motion takes place in the (x,z)-plane, where Oz points vertically upwards.

If the projection speed u is fixed, show that the particle can be made to pass through the point (a, b) for some choice of α if (a, b) lies below the parabola

z = (u²/2g) (1 - (g²x²/u⁴))

This is called the parabola of safety. Points above the parabola are ‘safe' from the projectile. An artillery shell explodes on the ground throwing shrapnel in all directions with speeds of up to 30 m s^-1. A man is standing at an open window 20 m above the ground in a building 60 m from the blast. Is he safe? [Take g = 10 m s^-2.]