The 1D velocity of an animated space alien is well-represented by the expression v(t)= t3 -11t2 +38t -40 with t in seconds and v in m/sec. Assume that x(1)=0 m and consider the motion between t = 1 and t = 6 sec. (a) What are the (implied) units of the number 11 in the expression for v(t)? (b) During what times is the alien moving in the positive direction? At what times is the alien at rest? (c) What is the average acceleration of the alien between 2 and 5 sec? (d) What is the instantaneous acceleration of the alien at t = 5 sec? (e) What is the average velocity of the alien between 1 and 5 sec (f) Where is the alien at t = 4 sec? (x = ?)
In: Physics
5. Equations of the form y’ = P(x)*y^2 + Q(x)*y + R(x) are called Riccati equations.
i) If we know a solution y = φ(x) of this equation, then any other solution
can be written in the form y(x) = φ(x)+ 1/v(x), where v(x) is an unknown
function which satisfies a certain linear equation. Using the fact that
φ and y both solve the above Riccati equation, find the differential
equation that v satisfies.
ii) Consider the equation 3y’ + y^2 +2/(x^2) = 0. Find one solution of this equation by inspection.
iii) Use the method of part(i) to find the general solution of the equation
in (ii).
In: Math
A skydiver, weighing 180 lb (including equipment) falls vertically downward from an altitude of 4000 ft and opens the parachute after 10 s of free fall. Assume that the force of air resistance, which is directed opposite to the velocity, is of magnitude 0.75|v| when the parachute is closed and is of magnitude 10|v| when the parachute is open, where the velocity v is measured in ft/s. (A computer algebra system is recommended. Use g = 32 ft/s2 for the acceleration due to gravity. Round your answers to two decimal places.)
Find the distance fallen before the parachute opens.
What is the limiting velocity vLafter the parachute opens?
Determine how long the sky diver is in the air after the parachute opens.
In: Advanced Math