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?

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To derive the equation for the trajectory of the probe during the time the rocket is fired, you can use the concept of Newton's Second Law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration. In this case, the net force acting on the probe is due to the thrust provided by the rocket.

First, let's break down the given information:

Mass of the space probe, m = 2000 kg
Initial velocity of the space probe along the x-axis, Vx = 12 m/s
Thrust provided by the rocket along the y-axis, F = 1800 N
Duration of the rocket firing, t = 1.5 s

Now, let's calculate the acceleration of the probe along the y-axis using Newton's Second Law:

F = ma
1800 N = (2000 kg)a
a = 1800 N / 2000 kg
a = 0.9 m/s^2

Since the probe is initially moving only along the x-axis and the thrust is provided only along the y-axis, the acceleration along the x-axis remains zero. Therefore, the velocity of the probe along the x-axis remains constant at 12 m/s.

Now, to derive the equation for the trajectory of the probe during the rocket firing, we need to find the displacement along the y-axis (denoted by y) as a function of the displacement along the x-axis (denoted by x). This can be done by calculating the area under the acceleration-time graph during the rocket firing.

Since the acceleration is constant, the area under the acceleration-time graph is given by the formula:

Area = a * t

Plugging in the values, we get:

Area = 0.9 m/s^2 * 1.5 s
Area = 1.35 m/s

This area represents the change in velocity along the y-axis during the rocket firing. Since the initial velocity along the y-axis is zero, the final velocity along the y-axis at the end of 1.5 s is:

Vy = Area / t
Vy = 1.35 m/s / 1.5 s
Vy = 0.9 m/s

Now, let's calculate the final speed of the probe at the end of 1.5 s by calculating the magnitude of the final velocity vector:

v = sqrt(Vx^2 + Vy^2)
v = sqrt((12 m/s)^2 + (0.9 m/s)^2)
v = sqrt(144 m^2/s^2 + 0.81 m^2/s^2)
v = sqrt(144.81 m^2/s^2)
v ≈ 12.04 m/s

Therefore, the final speed of the probe at the end of 1.5 s is approximately 12.04 m/s.