A constant force, acting for 3.6 ×10^4 s(10 h), brings a spaceship of mass 2200 kg from rest

step 1 Question A, one that's determining the force. When the force can be equal to the momentum divide

A constant force, acting for 3.6 ×10^4 s(10 h), brings a...

A constant force, acting for $3.6 \times 10^{4} \mathrm{~s}(10 \mathrm{~h})$, brings a spaceship of mass $2200 \mathrm{~kg}$ from rest to speed $0.70 c$. (a) What is the magnitude of the force? (b) What is the initial acceleration of the spaceship? Comment on the magnitude of the answer.

Key Concepts

Newton's Second Law in a Relativistic Context

This concept involves understanding that force is related to the time rate of change of momentum. In classical mechanics, one often writes F = ma; however, at speeds approaching the speed of light, momentum must be defined relativistically (p = ?mv), and thus the relation becomes F = d(?mv)/dt. This highlights that even with a constant applied force, the acceleration does not remain constant because the factor ? (the Lorentz factor) increases with speed.

Relativistic Momentum

At speeds near the speed of light, the momentum of an object is given by p = ?mv, where the Lorentz factor ? accounts for time dilation effects and is defined as 1/?(1 - v²/c²). This modification ensures that as an object’s speed approaches the speed limit in the universe (c), its momentum increases more rapidly than predicted by the classical relation, fundamentally altering the dynamics under a given force.

Impulse-Momentum Theorem

This theorem states that the impulse (the product of force and the time interval over which it is applied) equals the change in momentum of the object. In practice, if a constant force is applied over a known time period, one can determine the momentum change, and by using the relativistic definition of momentum, deduce the magnitude of the force necessary to achieve a specified change in velocity.

Coordinate vs. Proper Acceleration

In special relativity, the acceleration measured in the inertial frame (coordinate acceleration) differs from the acceleration that would be experienced by occupants of the accelerating spaceship (proper acceleration). While the initial coordinate acceleration might be calculated straightforwardly using F = ma at low speeds, as the velocity approaches relativistic speeds, the effective coordinate acceleration decreases even under a constant force. This distinction is crucial for understanding the dynamic limits imposed by relativity.

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