Question

Consider a \(10^{4} \mathrm{~kg}\) spaceship in deep space. Suppose the spaceship's thruster produces \(100 \mathrm{~N}\) of constant thrust. If the spaceship is moving at \(50 \mathrm{~m} / \mathrm{s}\) and the thruster is anti- aligned with the spaceship's velocity, how long should the thruster fire to bring the spaceship to rest with respect to absolute space?

Short answer

The thruster should fire for 5000 seconds.

Step-by-step solution

Identify the Known Quantities

First, note down what is given in the problem: \( m = 10^4 \text{ kg} \) (mass of the spaceship), \( F = 100 \text{ N} \) (force of the thruster), \( v_i = 50 \text{ m/s} \) (initial velocity of the spaceship), and \( v_f = 0 \text{ m/s} \) (final velocity).

Understand the Relationship

The relationship we use is Newton’s second law and the definition of acceleration, connecting force, mass, and acceleration: \( F = m \cdot a \). This can be rearranged to find acceleration as \( a = \frac{F}{m} \).

Calculate the Acceleration

Determine the acceleration of the spaceship by plugging in the values of force and mass: \( a = \frac{100 \text{ N}}{10^4 \text{ kg}} = 0.01 \text{ m/s}^2 \). Note that the acceleration is negative since the thrust is anti-aligned with the velocity.

Use Kinematics to Find Time

Use the kinematic equation \( v_f = v_i + a \cdot t \) to solve for time \( t \). Substituting the known values: \( 0 = 50 + (-0.01) \cdot t \).

Solve for Time

Rearrange the equation to solve for time: \( t = \frac{-50}{-0.01} = 5000 \text{ s} \).

Conclusion

The thruster needs to be fired for \(5000\) seconds to bring the spaceship to rest in absolute space.

Key concepts

Kinematics

Kinematics is a branch of mechanics that describes the motion of objects without considering the forces that cause this motion. In our context, it helps us understand how an object like a spaceship changes its velocity over time. The primary kinematic equation used is:\[ v_f = v_i + a \cdot t \]Where:

• \( v_f \): Final velocity
• \( v_i \): Initial velocity
• \( a \): Acceleration
• \( t \): Time

Kinematics allows us to find the time it takes for the spaceship to stop. Here, the initial velocity \( v_i \) of the spaceship is \( 50 \, \text{m/s} \), aiming for a final velocity \( v_f \) of \( 0 \, \text{m/s} \). By understanding the acceleration, we can calculate the duration the thrusters must fire to achieve this change.

Force and Acceleration

Force and acceleration are critically linked through Newton's Second Law of Motion. This law states that the force acting on an object is equal to its mass multiplied by its acceleration. It provides a fundamental method to calculate acceleration from a known force:\[ F = m \cdot a \]Rearranging gives us the acceleration formula:\[ a = \frac{F}{m} \]For the spaceship in the problem:

• \( F = 100 \, \text{N} \)
• \( m = 10^4 \, \text{kg} \)

Thus, the absolute acceleration is \( 0.01 \, \text{m/s}^2 \). It's critical to note that if force direction opposes velocity, it results in negative acceleration (deceleration), slowing the object down.In this case, the thruster force is directly opposing the spaceship's movement, resulting in a negative acceleration until it halts.

Spaceship Dynamics

Spaceship dynamics is an exciting realm of physics as it involves understanding how forces, motion, and dynamics manifest within space's vacuum environment. ### Understanding Spaceship Dynamics 1. **Mass and Thrust**: Spacecrafts have significant mass, and overcoming their inertia requires an external force, like thrust from engines. 2. **No Friction**: In deep space, there is no frictional force opposing the spaceship's motion, making it solely dependent on thrust for acceleration or deceleration. ### Bringing a Spaceship to Rest In bringing the spaceship to rest:

• The thrust is precisely calibrated to oppose the initial velocity of the craft.
• Newton's laws become essential in powering these dynamics, dictating the spacial velocity changes.
• The calculated acceleration and kinematic equations provide the necessary duration of thrust application to slow down the spaceship.

These elements work together to solve the problem of stopping a moving object in space, outlining how space travel navigates Newtonian physics.