Question

A \(2500-\mathrm{kg}\) unmanned space probe is moving in a straight line at a constant speed of \(300 \mathrm{~m} / \mathrm{s}\). A rocket engine on the space probe executes a burn in which a thrust of \(3000 \mathrm{~N}\) acts for \(65.0 \mathrm{~s}\). What is the change in momentum (magnitude only) of the probe if the thrust is backward, forward, or sideways? Assume that the mass of the ejected fuel is negligible compared to the mass of the space probe.

Short answer

The change in momentum of the probe is 195000 kg.m/s. Its orientation can be positive, negative, or sideways depending on the direction of the applied force.

Step-by-step solution

Understanding the Impulse-Momentum Theorem

The Impulse-Momentum Theorem states that the change in momentum of an object equals the impulse applied to it. Mathematically, this can be represented as \(\Delta p = F \cdot \Delta t\), where \(\Delta p\) is the change in momentum, \(F\) is the force applied and \(\Delta t\) is the time duration that the force is applied.

Calculate the Change in Momentum

With a force of \(3000 N\) acting on the probe for a duration of \(65.0 s\), we calculate the change in momentum: \(\Delta p = F \cdot \Delta t = 3000 N \cdot 65.0 s = 195000 kg \cdot m/s\).

Discuss the Direction

If the thrust of force is exerted backward, the change in direct momentum ups the vehicle's speed making the change in momentum positive (+195000 kg.m/s). If, on the other hand, it's exerted forwards, it slows the vehicle down therefore providing a resultant negative change (-195000 kg.m/s). A sideways thrust changes the object’s direction but not its speed therefore no influence over its magnitude; the change remains \(195000 kg.m/s\), but with a direction that is perpendicular to the original motion pathway.

Key concepts

Understanding Change in Momentum

When discussing the concept of momentum in physics, it's important to understand that momentum is a vector quantity, meaning it has both magnitude and direction. It is defined as the product of an object's mass and its velocity (\( p = m \times v \)).

The change in momentum, denoted as \( \Delta p \), occurs when an object experiences a variation in either speed, direction, or both. This change can be a result of various forces acting upon the object for a certain period. According to the impulse-momentum theorem, this change is equal to the impulse applied, which is the product of the force (\( F \)) and the time span (\( \Delta t \)) during which the force acts (\( \Delta p = F \times \Delta t \)).

• For a constant force applied over time, impulse is straightforward to compute, and the total change in momentum can be determined as illustrated in the space probe exercise.
• If either the force or time is variable, more complex calculus may be needed to determine the change in momentum.

In our space probe example, the force applied is constant, and regardless of direction, the magnitude of change in momentum remains the same—195,000 kg·m/s. However, the direction of this change is contingent on the direction of the force, denoting the vector nature of momentum.

Deciphering Thrust Force

Thrust force is a specific type of force commonly discussed in the context of rockets, jet engines, or any system that propels an object through a fluid medium like air or space. Thrust is generated by the expulsion or acceleration of mass in one direction, producing a reaction force in the opposite direction—famously explained by Newton’s third law of motion (for every action, there's an equal and opposite reaction).

When we apply thrust to an object:

• It either accelerates the object if it's in the direction of the current velocity,
• Decelerates the object if it's in the opposite direction, or
• Changes the object's direction if applied perpendicularly to the velocity.

In the case of the space probe, a backward thrust increases the probe's speed because it adds to its momentum in the direction it's already moving. A forward thrust would decrease its speed, acting contrary to the probe's momentum, while a sideways thrust would alter the probe's direction without changing the magnitude of its momentum (although it pivots the momentum vector).

The calculation of the space probe’s change in momentum due to thrust force doesn't consider the ejected fuel mass, assuming it's negligible. This is a simplification often made in physics problems to focus on essential concepts without getting mired in overly complex calculations.

Exploring Momentum Conservation

Momentum conservation is a fundamental principle in physics, stating that the total momentum of a closed system remains constant if no external forces are acting on it. This principle is derived from Newton's first law and is evident when observing isolated systems.

Several key points about momentum conservation are:

• It holds true for both linear and angular momentum,
• It's an integral part of analyzing collisions and explosions where no external forces impact the total system, and
• It suggests that the center of mass for a system of particles moves with a constant velocity unless acted upon by an external force.

We see this conservation at play when considering our space probe example as well. If the space probe were part of a system where the fuel is considered, the momentum of the probe and the ejected fuel together would remain the same before and after the burn. However, because the exercise assumes the mass of ejected fuel to be negligible, we look only at the probe’s momentum change, which simplifies our analysis and clearly depicts how external forces—like thrust—affect a single object's momentum.

In any scenario where external forces come into play—like our space probe acted upon by thrust force—we must carefully consider the system's boundaries and the internal versus external interactions to accurately apply the principle of momentum conservation.