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?

Thrust is in force units. For all intents and purposes, there is a force in the y-direction of 1800 N for 1.5 seconds.

The motions in the x- and y-directions being linearly independent, we can process them separately and relate the motion using the parameter t=time.

vx(t)=12 m/s (constant)
force in y-direction, F = 1800 N
mass, m = 2000 kg
Since Force F=ma, where a = acceleration
acceleration=F/m=1800/2000 m/s²
=0.9 m/s²
Initial velocity in the y-direction = 0
therefore
vy(t)=0+at=0.9t (0≤t≤1.5)

The trajectory is given by the parametric formulae vx(t) and vy(t) in terms of time, t.

The speed at any time t within the firing period (1.5 s) can be obtained by the vectorial sum of the two velocities, vx(t) and vy(t):

Speed for 0≤t≤1.5
S(t)=sqrt(vx(t)²+vy(t)²)
=sqrt(12²+(0.9t)²)

Speed at the end of the rocket firing period
=S(1.5)

To derive the equation for the trajectory of the probe during the time the rocket is fired, you can use the laws of motion and apply them to this specific scenario.

1. Determine the initial conditions:
- Mass of the space probe: 2000 kg
- Initial velocity along the x-axis: 12 m/s
- Thrust provided by the rocket along the y-axis: 1800 N
- Duration of rocket firing: 1.5 s

2. Determine the net force acting on the probe during this time:
The net force (F_net) acting on the probe is the vector sum of the forces acting in the x and y directions.
- The force along the x-axis is zero because there are no external forces acting on the probe in the x-direction.
- The force along the y-axis is the thrust provided by the rocket, which is 1800 N.

3. Apply Newton's second law of motion:
F_net = m * a
- Since we know the force in the y-direction (1800 N), we can write:
1800 N = m * a_y

4. Determine the acceleration in the y-direction:
- We can use the formula:
a_y = F_net / m
a_y = 1800 N / 2000 kg
a_y = 0.9 m/s^2

5. Integrating the acceleration in the y-direction to get the velocity:
- Since the acceleration is constant during the 1.5 s, we can integrate it to find the change in velocity:
Δv_y = a_y * t
Δv_y = 0.9 m/s^2 * 1.5 s
Δv_y = 1.35 m/s

6. Calculating the final velocity of the probe in the y-direction:
- The final velocity in the y-direction is the sum of the initial velocity (0 m/s) and the change in velocity:
v_yf = v_yi + Δv_y
v_yf = 0 m/s + 1.35 m/s
v_yf = 1.35 m/s

7. Deriving the equation for the trajectory:
- The equation for the trajectory can be determined using the equations of motion. Since the probe only experiences acceleration in the y-direction, the equation becomes:
y = v_yi * t + (1/2) * a_y * t^2
As the initial velocity along the y-axis is zero, we can simplify the equation:
y = (1/2) * a_y * t^2
y = (1/2) * 0.9 m/s^2 * (1.5 s)^2
y = 1.0125 m

The equation for the trajectory of the probe during the time the rocket is fired is y = 1.0125 m, assuming the initial position is taken as zero.

To compute the final speed of the rocket at the end of the 1.5 s, we can use the equation:
- The final velocity of the rocket can be determined by adding the change in velocity in the y-direction (1.35 m/s) to the initial velocity (0 m/s):
v_rf = v_ri + Δv_y
v_rf = 0 m/s + 1.35 m/s
v_rf = 1.35 m/s

Therefore, the final speed of the rocket at the end of the 1.5 s is 1.35 m/s.