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You and your crew must dock your \(25000 \mathrm{kg}\) spaceship at Spaceport Alpha, which is orbiting Mars. In the process, Alpha's control tower has requested that you ram another vessel, a freight ship of mass \(16500 \mathrm{kg},\) latch onto it, and use your combined momentum to bring it into dock. The freight ship is not moving with respect to the colossal Spaceport Alpha, which has a mass of \(1.85 \times 10^{7} \mathrm{kg} .\) Alpha's automated system that guides incoming spacecraft into dock requires that the incoming speed is less than \(2.0 \mathrm{m} / \mathrm{s}\). (a) Assuming a perfectly linear alignment of your ship's velocity vector with the freight ship (which is stationary with respect to Alpha) and Alpha's docking port, what must be your ship's speed (before colliding with the freight ship) so that the combination of the freight ship and your ship arrives at Alpha's docking port with a speed of \(1.50 \mathrm{m} / \mathrm{s} ?\) (b) How does the velocity of Spaceport Alpha change when the combination of your vessel and the freight ship successfully docks with it? (c) Suppose you made a mistake while maneuvering your vessel in an attempt to ram the freight ship and, rather than latching on to it and making a perfectly inelastic collision, you strike it and knock it in the direction of the spaceport with a perfectly elastic collision. What is the speed of the freight ship in that case (assuming your ship had the same initial velocity as that calculated in part (a))?

Step by step solution

01

Understanding the Problem

We need to dock a spaceship at Spaceport Alpha with an incoming speed of less than 2 m/s after ramming and attaching to a freight ship. Initially, only our spaceship is moving, and after collision, the combined system must move at 1.5 m/s towards the spaceport.

02

Conservation of Momentum for Part (a)

We use the principle of conservation of momentum for part (a). The initial momentum is the mass of our spaceship times its initial velocity (\(v_i\)). The final momentum is the combined mass of our spaceship and the freight ship times their final velocity (1.5 m/s). Set the initial and final momenta equal to solve for \(v_i\). The equation is: \[m_s v_i = (m_s + m_f) v_f\]Plug in the values: \[25000 \times v_i = (25000 + 16500) \times 1.5\] Solve for \(v_i\).

03

Calculating Initial Speed of Ship (a)

Calculate \(v_i\) using the equation: \[v_i = \frac{(25000 + 16500) \times 1.5}{25000}\] Perform the calculation to find the speed before collision.

04

Conservation of Momentum for Part (b)

To find how Spaceport Alpha's speed changes, apply conservation of momentum to the docking process. Initially, the docked ship and freight ship are moving, and Alpha is stationary. After docking, all three are combined. The equation is: \[(m_s + m_f) v_f = (m_s + m_f + m_a) v_{final-alpha}\]Since initially, Alpha's speed is 0, the final speed change for Alpha depends on the final system's momentum.

05

Calculating Alpha's Speed Change (b)

Since Alpha's mass is significantly larger than the docked system: \[v_{final-alpha} \approx \frac{(m_s + m_f) v_f}{m_a}\] Where \(m_a = 1.85 \times 10^7\) kg. Compute this to show that Alpha's speed change is negligible.

06

Dynamics of a Perfectly Elastic Collision (c)

For a perfectly elastic collision, both momentum and kinetic energy are conserved. First, apply the conservation of momentum: \[m_s v_i = m_s v_s + m_f v_f\] Next, apply the conservation of kinetic energy: \[\frac{1}{2}m_s v_i^2 = \frac{1}{2}m_s v_s^2 + \frac{1}{2}m_f v_f^2\] Solve these equations to find the ejection speed \(v_f\) of the freight ship after collision.

07

Solving for Freight Ship Speed (c)

Using the kinematic equations and the value for \(v_i\) found earlier, solve the equations to find the freighter's final speed. This involves solving simultaneous equations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inelastic Collision

Inelastic collisions are a specific type of collision where the colliding bodies stick together after impact. During such events, kinetic energy is not conserved, although the total momentum remains conserved.
In the context of our spacecraft problem, we are involving the docking of two ships by ramming into one another and creating a perfectly inelastic collision.

• The spaceship, upon impact, latches onto the freight ship creating one single mass unit.
• The momentum before the collision equals the momentum after the collision; however, some kinetic energy is converted to other forms like sound or heat.
• To calculate the necessary speed of the spaceship before the collision, conservation of momentum is applied, merging the two ships’ masses to attain a combined speed of 1.5 m/s after the collision.

In practical scenarios like this space docking, ensuring inelastic collisions can simplify dock operations by keeping vessels attached. Yet, it’s important to note that not all kinetic energy stays in a useful mechanical form, reducing overall energy efficiency in the system.

Elastic Collision

An elastic collision contrasts an inelastic collision because it involves both momentum and kinetic energy conservation. After an elastic collision, neither object sticks together; they simply bounce off each other.
In our spaceship scenario, a mistake during docking causes the collision to be perfectly elastic.

• For this scenario, both momentum and kinetic energy equations must be used to understand post-collision velocities.
• The spaceship losses some of its momentum, but the resulting freight ship gains velocity according to the conservation laws.
• Simultaneous equations derived from both conserved momentum and conserved kinetic energy provide the freight ship's speed due to this elastic collision.

The complexity of solving an elastic collision makes it interesting yet challenging. It’s crucial during planning to consider if the colliding bodies will separate or stick, as efficiency and behavior alter significantly post-collision.

Kinetic Energy

Kinetic energy refers to the energy of motion, calculated as \ \( KE = \frac{1}{2}mv^2 \ \). In both elastic and inelastic collisions, kinetic energy serves an important role in understanding the dynamics of moving bodies.
In the problem of docking spacecraft, consideration of kinetic energy helps differentiate between inelastic and elastic collisions.

• In inelastic collisions, kinetic energy is not conserved. It's transformed into other energy forms, reducing the energy available in motion.
• However, in elastic collisions, kinetic energy before and after is the same, preserving the motions of the striking bodies.
• When analyzing these scenarios, it's important to account for energy losses or distributions as they can affect the outcome efficiency and performance.

The transition and conservation of kinetic energy between moving vessels reveal the mechanical interactions and result in either a joined mass or separated forms, revealing crucial insights into the energy dynamic during space missions.