These look like pretty standard problems. What is giving you trouble? Which parts have you tried, and with what results
You know what the graphs of e^-x and ln(x) look like, so start from there.
acceleration is just dv/dt, so start there.
QA: as x goes to infinity,
i) describe how 1-exp(-x) behaves
ii) describe how its derivative behaves
iii) describe how ln(x) behaves
iv) describe how the derivative of ln(x) behaves
v) Can we say "If the derivative goes to zero, then the original function stays bounded" as x goes to infinity? Explain.
QB: The Rocket Equation
The velocity of a rocket t seconds after liftoff from earth can be modeled by
v(t) = -g*t - v_e * Ln( (m-r*t)/m )
where
g = 9.8 m/s^2, the usual earth gravity value,
v_e = exhaust velocity, 3000 m/s (the e here isn't related to 2.71828...)
m = initial mass of the rocket+fuel: 30,000 kg
r = rate of using fuel = 160 kg/s
i) Find a formula for the acceleration function a(t) and graph it, t=0 to 60. Do the formula work by hand, but not the graphing, of course.
ii) Find a formula for the jerk function j(t) = a'(t) and graph it, t=0 to 60. Do the formula work by hand, but not the graphing.
iii) optional: what happens as t approaches 187.5 ? Explain. Hint: it doesn’t have anything to do with breaking free of gravity or of the atmosphere. Think about the “m” and “r” parameters
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