A Federation starship (2500000 kg) uses its tractor beam to pull a shuttlecraft (10000) aboard from a distance of 7 km away. The tractor beam exerts a constant force of 25000 N on the shuttlecraft. Both spacecraft are initially at rest. How far does the starship move (in m) as it pulls the shuttlecraft aboard?

Well, let's calculate it step by step. First, we need to determine the total work done by the tractor beam on the shuttlecraft. We can use the work-energy theorem, which states that the work done on an object is equal to its change in kinetic energy.

Since the shuttlecraft starts from rest, its initial kinetic energy is zero. The final kinetic energy is given by (1/2)mv^2, where m is the mass of the shuttlecraft and v is its final velocity.

The work done on the shuttlecraft is equal to the force applied multiplied by the distance over which it is applied. In this case, the distance is 7 km, which we'll convert to meters (7000 m).

So, the work done on the shuttlecraft is given by:

Work = force * distance
= 25000 N * 7000 m
= 175,000,000 Nm

Now, we equate the work done on the shuttlecraft to its change in kinetic energy:

175,000,000 Nm = (1/2) * 10,000 kg * v^2

Now, let's solve the equation for v:

v^2 = (2 * 175,000,000 Nm) / 10,000 kg
v^2 = 35,000 m^2/s^2

Taking the square root of both sides, we get:

v = √(35,000 m^2/s^2)
v ≈ 187.082 m/s

Once we know the final velocity of the shuttlecraft, we can calculate the distance the starship moves using the equation:

Distance = velocity * time

Since the starship and shuttlecraft start from rest and the force applied by the tractor beam is constant, the time it takes for the shuttlecraft to reach its final velocity is the same as the time it takes for the starship to move the same distance. So, the distance the starship moves is:

Distance = 187.082 m/s * 7000 m
Distance ≈ 1,309,574.4 m

So, the starship moves approximately 1,309,574.4 meters as it pulls the shuttlecraft aboard.

But hey, let's hope the starship captain isn't in a rush, or else they might say, "Beam me up, Scotty, and hurry!"

To solve this problem, we can use the equations of motion. The equation that relates force, mass, and acceleration is:

F = m * a

Where F is the force exerted on an object, m is the mass of the object, and a is the acceleration.

We can rearrange this equation to solve for acceleration:

a = F / m

In this case, the force exerted by the tractor beam is 25000 N, and the mass of the shuttlecraft is 10000 kg. So, the acceleration of the shuttlecraft can be calculated as:

a = 25000 N / 10000 kg = 2.5 m/s^2

Since the initial velocity of both spacecraft is zero (since they are at rest), we can use the following equation to calculate the displacement of an object with constant acceleration:

s = ut + (1/2) * a * t^2

Where s is the displacement, u is the initial velocity, t is the time, and a is the acceleration.

In this case, we know that the displacement is 7 km (7000 m) and the acceleration is 2.5 m/s^2. We want to find the time taken for the shuttlecraft to be pulled onboard, so we rearrange the equation as follows:

t^2 = (2s) / a

t^2 = (2 * 7000 m) / (2.5 m/s^2)
t^2 = 5600 s^2

Taking the square root of both sides, we find:

t = √(5600 s^2)
t = 74.83 s

Now that we know the time taken for the shuttlecraft to be pulled onboard, we can use the equation:

s = ut + (1/2) * a * t^2

Since the initial velocity of the starship is zero, the equation simplifies to:

s = (1/2) * a * t^2

s = (1/2) * 2.5 m/s^2 * (74.83 s)^2
s = 2.415 * 10^4 m

Therefore, the starship moves a distance of 24,150 meters (or 24.15 km) as it pulls the shuttlecraft aboard.