Define work and power, and shew that, when a force \(F\) is moving its point of application with velocity \(v\), the power is measured by \(Fv\). An engine of 300 horse-power pulls a train of 200 tons mass up an incline of 1 in 120, the resistance of wind and rails being 10 lb. weight per ton. Find the maximum velocity acquired, correct to one place of decimals in miles per hour.
Prove that the path of a particle projected from a given point with a given velocity is a parabola, and that the velocity at any point is equal to the velocity acquired by a particle falling freely from the directrix to that point. A particle is projected from a point at a height \(h\) above a horizontal plane with velocity \(\sqrt{gh}\); shew that the farthest point in the plane which the particle can reach is at a distance \(2h\) from the point of projection.