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, we can use the principles of Newton's laws of motion and kinematics.

First, let's analyze the forces acting on the probe. Before the rocket fires, the only force acting on the probe is its weight, which we can denote as Fg. After the rocket fires, an additional force from the thrust is added, which we can denote as Ft.

Since the probe is initially moving along the x-axis and the rocket's thrust is along the y-axis, the rocket's thrust will not directly affect the probe's motion in the x-direction. Therefore, we can consider the motion in the x-direction to be unaffected by the rocket's thrust.

In the y-direction, the net force acting on the probe during the time the rocket is fired is the sum of the gravitational force and the force from the rocket's thrust:

F_net = Fg + Ft

The gravitational force can be calculated using the formula Fg = mg, where m is the mass of the probe (2000 kg) and g is the acceleration due to gravity. Assuming g is approximately 9.81 m/s^2, we can calculate Fg:

Fg = (2000 kg) * (9.81 m/s^2)

Next, we need to calculate the force from the rocket's thrust. We're given that the thrust is 1800 N and that it acts for 1.5 seconds:

Ft = (1800 N) * (1.5 s)

Now that we have both Fg and Ft, we can calculate the net force:

F_net = Fg + Ft

Now, we can apply Newton's second law of motion, which states that the net force on an object is equal to the product of its mass and acceleration:

F_net = m * a

Since the net force is in the y-direction, and assuming there is no air resistance or other external forces, the acceleration will be in the y-direction as well.

With this information, we can now create an equation for the trajectory of the probe during the time the rocket is fired, using the formula y = a function of x.

To compute the final speed of the rocket at the end of the 1.5 seconds, we can use the formula for final velocity:

vf = vi + at

Where vf is the final velocity, vi is the initial velocity (12 m/s), a is the acceleration due to the rocket's thrust, and t is the time the rocket is fired (1.5 seconds).

Plug in the given values to compute the final speed of the rocket.

To derive the equation for the trajectory of the probe during the time the rocket is fired, we can use the equations of motion.

First, let's assume that the initial position of the probe is x = 0 and y = 0.

During the time the rocket is fired, the only force acting on the probe is the thrust of the rocket in the y direction.

The net force on the probe is given by the equation:

F_net = m * a

where F_net is the net force, m is the mass of the probe, and a is the acceleration.

Since there are no other forces acting on the probe, the net force is equal to the thrust of the rocket:

F_net = 1800 N

We can rearrange the equation to solve for the acceleration:

a = F_net / m = 1800 N / 2000 kg = 0.9 m/s^2

Now, let's derive the equation for the trajectory of the probe during the time the rocket is fired using the formula y = a function of x.

We know that the acceleration in the y direction is constant, so we can use the kinematic equation:

y = y_0 + v_0y * t + (1/2) * a_y * t^2

where y is the vertical position, y_0 is the initial vertical position (0 in this case), v_0y is the initial vertical velocity (0 in this case), t is the time (1.5 seconds), and a_y is the acceleration in the y direction (0.9 m/s^2).

Simplifying the equation, we get:

y = (1/2) * a_y * t^2

y = (1/2) * 0.9 m/s^2 * (1.5 s)^2

y = 0.608 m

The equation for the trajectory of the probe during the time the rocket is fired is y = 0.608 m.

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

v = v_0 + a * t

where v is the final velocity, v_0 is the initial velocity (0 in this case), a is the acceleration (0.9 m/s^2), and t is the time (1.5 seconds).

Simplifying the equation, we get:

v = a * t

v = 0.9 m/s^2 * 1.5 s

v = 1.35 m/s

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