Question

You and your team launch a rocket from a 5-foot platform with an initial vertical velocity of 100 ft. The only force that acts on the rocket after the initial launch is gravity.

A) If the equation for the height of the rocket is h = a(t)? + b(t) + c, where h is height in feet and r is time in seconds, what are the values of a, b and c for this problem? (Hint: The force of gravity is -16 feet per second squared. Really it's -32 feet per second squared but then you have to divide by two because calculus)

B) At what time will the rocket reach maximum height? What is its maximum height?

C) How long after the launch will the rocket hit the ground?

D) Your team decides the rocket is descending too quickly and decides to attach a parachute to open after a certain number of seconds. You make a parachute that causes the rocket to descend at a steady 5 feet per second, so after
the parachute opens, the rocket should descend according to the function y = -5x + b. If you want the entire trip of the
rocket to take 9 seconds, what is the value of b and when should your rocket open it's parachute?

Answer 1

A) The equation for the height of the rocket is h = a(t)^2 + b(t) + c, where h is height in feet and t is time in seconds. Since the rocket is launched with an initial vertical velocity of 100 ft, the value of a is 1/2*(-16) = -8. The rocket is launched from a 5-foot platform, so the initial height of the rocket is 5 feet. Therefore, the value of c is 5. To find the value of b, we need to use the initial velocity of the rocket. At t=0, the initial height is 5 feet and the initial velocity is 100 feet per second. Thus, b = 100t + 5.

B) To find the time when the rocket reaches maximum height, we need to find the vertex of the parabolic equation. The vertex of the parabola is given by the formula t = -b/2a. Plugging in the values of a and b, we get t = -100/-16 = 6.25 seconds. To find the maximum height, we need to plug in this value of t into the equation for h: h = -8(6.25)^2 + 100(6.25) + 5 = 320.3125 feet.

C) To find the time when the rocket hits the ground, we need to find the time when h = 0. Setting h to 0 in the equation, we get: 0 = -8t^2 + 100t + 5. Using the quadratic formula, we get t = (-100 +/- sqrt(100^2 - 4*(-8)5))/(2(-8)) = 12.81 seconds. Therefore, the rocket hits the ground after 12.81 seconds.

D) The equation for the height of the rocket after the parachute opens is y = -5x + b. We want the entire trip to take 9 seconds, so the rocket will be descending for (9-6.25) = 2.75 seconds. During this time, the rocket will descend a total of 2.75*5 = 13.75 feet. Since the rocket was at a height of 320.3125 feet when the parachute opened, it needs to descend a further 13.75 feet to reach the ground. Therefore, the value of b in the equation for y is 320.3125 + 13.75 = 334.0625. To find the time when the parachute should open, we need to solve the equation h = -5t + 334.0625 for t, where h is the height of the rocket. Setting h to 100 feet (the height at which the parachute should open), we get: 100 = -5t + 334.0625. Solving for t, we get t = 46.8125 seconds. Therefore, the parachute should open after 46.8125 - 6.25 = 40.5625 seconds.