A 2000 kg space probe is moving rightward in empty space along the x axis at 12 m/s. One of the probe’s rockets is fired providing a thrust of 1800j N along the y axis. The rocket fires for 1.5 s.

How can I derive the equation for the trajectory of the probe during the time the rocket is fired using the formula y = a function of x and compute the final speed of the rocket at the end of the 1.5 s?

Write and solve separate differential equations for the x and y coordinates, during the rocket firing. You will end up with separate equations for x and y.

Once you have those functions, try to write y as a function of x. The Vx velocity component will remain 12 m/s.
That means x = 12 t, if position is measured from the time zero location.

Then derive a separate equation for Vy = dy/dt

The speed is sqrt[Vx^2 + Vy^2] = sqrt[144 + Vy^2]

To derive the equation for the trajectory of the probe during the time the rocket is fired, you can use the concept of linear motion.

Since the probe is moving only along the x-axis initially, we can assume that the motion along the y-axis is independent of the motion along the x-axis. Therefore, the motion along the x-axis remains constant, and we only need to determine the motion along the y-axis during the time the rocket is fired.

First, let's find the net force acting on the probe along the y-axis during the time the rocket is fired. The net force (F_net) consists of two components: the rocket thrust (F_thrust) and the weight of the probe (mg). The weight of the probe is given by mg = 2000 kg * 9.8 m/s^2 = 19600 N.

F_net = F_thrust - mg
F_net = 1800 N - 19600 N
F_net = -17800 N

The negative sign indicates that the net force is acting in the negative y-direction, opposite to the positive y-axis.

Now, using Newton's second law, which states that force (F_net) is equal to mass (m) multiplied by acceleration (a), we can relate the net force and the acceleration.

F_net = ma
-17800 N = (2000 kg) * a
a = -8.9 m/s²

The acceleration is negative because it is acting in the negative y-direction.

Now, we can use the equation of motion for constant acceleration to find the equation for the trajectory.

y = y_0 + v_0y * t + (1/2) * a * t²

Since the probe starts from rest along the y-axis, the initial velocity (v_0y) is 0 and the initial position (y_0) is also 0. Therefore, the equation simplifies to:

y = (1/2) * a * t²

Substituting the value of acceleration (a = -8.9 m/s²) and the time (t = 1.5 s) during which the rocket is fired:

y = (1/2) * (-8.9 m/s²) * (1.5 s)²
y = -18.9325 m

So, the equation for the trajectory of the probe during the time the rocket is fired is y = -18.9325 m.

To compute the final speed of the rocket at the end of the 1.5 s, we can use the equation of motion for constant acceleration:

v_f = v_0 + a * t

Since the initial velocity (v_0) along the y-axis is 0, the equation simplifies to:

v_f = a * t

Substituting the value of acceleration (a = -8.9 m/s²) and the time (t = 1.5 s):

v_f = (-8.9 m/s²) * (1.5 s)
v_f = -13.35 m/s

The final speed of the rocket at the end of the 1.5 s is -13.35 m/s along the negative y-axis.