Final answer:
a. The velocity as a function of time can be calculated by integrating the given components of acceleration with respect to time. b. The position as a function of time can be calculated by integrating the components of velocity with respect to time. c. The maximum height reached by the rocket can be determined by finding the time at which the vertical velocity component equals zero. d. The horizontal displacement when the rocket returns to y = 0 can be determined by finding the time at which the y-coordinate of position equals zero.
Step-by-step explanation:
a. Velocity as a function of time: To calculate the velocity as a function of time, we need to integrate the given components of acceleration with respect to time. Integrating ax with respect to time gives us vx = (0.8 m/s^4)t^3 + C1, where C1 is the constant of integration. Similarly, integrating ay with respect to time gives us vy = 8t - (0.5 m/s^3)t^2 + C2, where C2 is the constant of integration. The velocity as a function of time can then be represented as v = vxî + vyĵ.
b. Position as a function of time: To calculate the position as a function of time, we need to integrate the components of velocity with respect to time. Integrating the x-component of velocity, vx, gives us the x-coordinate of position, x = (0.2 m/s^4)t^4 + C1t + C3, where C3 is another constant of integration. Integrating the y-component of velocity, vy, gives us the y-coordinate of position, y = 4t^2 - (0.25 m/s^3)t^3 + C2t + C4, where C4 is another constant of integration. The position as a function of time can then be represented as ⃗r = xi + yĵ.
c. Maximum height reached by the rocket: The maximum height reached by the rocket can be determined by finding the time at which the vertical velocity component, vy, equals zero. Setting vy = 0 and solving for t, we get t = 8/3 s. Substituting this value into the equation for y, we get the maximum height as y = 4(8/3)^2 - (0.25 m/s^3)(8/3)^3 + C2(8/3) + C4.
d. Horizontal displacement when the rocket returns to y = 0: The horizontal displacement can be determined by finding the time at which the y-coordinate of position, y, equals zero. Setting y = 0 and solving for t, we get t = 0 or t = 8/3 s. The horizontal displacement is the difference between the x-coordinates of position at these two time points, Δx = [(0.2 m/s^4)(8/3)^4 + C1(8/3) + C3] - [(0.2 m/s^4)(0)^4 + C1(0) + C3].