Chapter 12.5, Problem 57E

Conservation of energy A projectile with mass m is launched into the air on a parabolic trajectory. For t ≥ 0, its horizontal and vertical coordinates are x(t) = u₀ t and $y\left(t\right)=-\frac{1}{2}g{t}^2+v_{0}t$, respectively. where u₀ is the initial horizontal velocity, v₀ is the initial vertical velocity, and g is the acceleration due to gravity. Recalling that $u\left(t\right)=x^{\prime }\left(t\right)$ and $v\left(t\right)=y^{\prime }\left(t\right)$ are the components of the velocity, the energy of the projectile (kinetic plus potential) is

     $E\left(t\right)=\frac{1}{2}m\left(u^2+v^2\right)+mgy.$

Use the Chain Rule to compute $E^{\prime }\left(t\right)$and show that $E^{\prime }\left(t\right)=0$, for all t ≥ 0. Interpret the result.