We can use the kinematic equations to find the position and velocity of the rocket at any time t, given its initial velocity and acceleration.
The position of the rocket at time t is given by:
x(t) = x0 + v0x t + 1/2 ax t^2
y(t) = y0 + v0y t + 1/2 ay t^2
where x0 and y0 are the initial position components (both zero in this case).
The velocity of the rocket at time t is given by:
vx(t) = v0x + ax t
vy(t) = v0y + ay t
At t = 0, the rocket is at the origin with velocity v0x = 1.00 m/s and v0y = 7.00 m/s. Therefore, we have:
x0 = 0, v0x = 1.00 m/s, and ax = 2.50 m/s^2
y0 = 0, v0y = 7.00 m/s, and ay = 9.00 - 1.40t m/s^2
(a) What is the maximum altitude reached by the rocket?
The maximum altitude will be reached when the vertical velocity of the rocket becomes zero, i.e., when vy = 0. We can use the equation for vy(t) to find the time t when this happens:
vy(t) = v0y + ay t = 0
t = -v0y/ay = -7.00 m/s / (9.00 - 1.40t) m/s^2
Substituting this value of t into the equation for y(t), we get the maximum altitude:
y_max = y(t) = 0 + 7.00 t - 1/2 ay t^2
y_max = (49/12) m
Therefore, the maximum altitude reached by the rocket is 4.08 m.
(b) What is the maximum distance from the origin attained by the rocket?
The maximum distance from the origin will be attained when the rocket returns to the ground, i.e., when y(t) = 0. We can use the equation for y(t) to find the time t when this happens:
y(t) = 0 = 7.00 t - 1/2 ay t^2
t = 0 or t = 14/9 s
The rocket returns to the ground at t = 14/9 s. Substituting this value of t into the equation for x(t), we get the maximum distance from the origin:
x_max = x(t) = 0 + v0x t + 1/2 ax t^2
x_max = (35/27) m
Therefore, the maximum distance from the origin attained by the rocket is 1.30 m.