CS 121 – Quiz 4 Questions 5 and 6

Question 5 We are given Blammo’s initial angle of elevation, initial velocity, and the height of the center of the ring. With this, we can define our xpos and ypos functions: – xpos := (t) -> v0 * cos(θ) * t – ypos := (t) -> v0 * sin(θ) * t – (1 / 2) * g * (t ^ 2) Don’t forget that θ must be in radians. It is given to us in degrees so we need to convert: – convert(51, units, degrees, radians)

We want Blammo to fly through the ring halfway through his flight, so we need to find when that is by finding his total flight time and dividing it in half: – peakTime := solve(ypos(t) = 0, t)[2] / 2 Solve will return a list of two solutions because Blammo not only lands on the ground, but also starts on the ground at t = 0. We are interested in the second solution. Also, notice that because we did not define v0, the answer for peakTime is given in terms of v0. This is what we want.

We can now find the y position in terms of v0, set it equal to the ring height, and solve for v0. We take the max because we want to ignore the negative root: – v := max(solve(ypos(peakTime) = ringHeight, v0)) We can then find the maximum distance by finding the final x position (at t = 2 * peakTime) in terms of v0, and plugging in the velocity we just calculated: – maxDistance := eval(xpos(2 * peakTime), v0 = v) And the answer: evalf([v, maxDistance])

Question 6 We are given the inclination angle (in degrees, don’t forget to convert to radians), the initial velocity (0), and the mass of the object. With this, we can define our v and t functions: – v := (t) -> v * sin(alpha) * t – d := (t) -> v0 * t + 16 * sin(alpha) * (t ^ 2) We can find the time it takes to travel the distance D to the bottom of the incline for part a: – maxTime := max(solve(d(t) = D, t)) And plug this into v for part b: – maxVel := v(maxTime)

For part c, we need to define and use the function K: – K := (m, v) -> m * (v ^ 2) – maxKin := K(m, maxVel) And finally, we need to convert to joules: – convert(maxKin, units, (pounds * feet ^ 2) / (seconds ^ 2), joules) For all of these parts, Maple TA expects an approximate answer, so don’t forget to use evalf.