Suppose a toy rocket is launched vertically into the air (at time \(t=0\)) and that the functions \(T(t)=2000/(t+10)\) and \(m(t)=100/(t+10)\) represent the thrust of the rocket (in N) and the mass (in kg) of the rocket respectively at \(t\) seconds for \(0\leq t\leq 10\). Assume air resists motion (in N) at a rate equal to \(10\) times the speed of the rocket in \(m/s\).

(a)
Let’s have \(v(t)\) be the vertical velocity of the rocket at \(0\leq t\leq 10\) seconds after launch. Let’s derive an IVP that \(v(t)\) satisfies using the most general form of Newton’s 2nd Law: \(\displaystyle \sum F=(m(t)v(t))'\) and \(g\approx 10\) meters per square second as the magnitude of the acceleration due to gravity.
(b)
Let’s solve this IVP in terms of an integral function \(I(t)=\displaystyle \int _{0}^{t}f(x)\;dx\).
(c)
Let’s determine the rocket’s approximate velocity at \(t=1\) and at \(t=10\) seconds (using a numerical approx. of the integral function).