Question

Both classically and relativistically, the force on an object is what causes a time rate of change of its momentum: \(F=d p / d t\). (a) Using the relativistically cotrect expression for momentum, show that $$ F=\gamma_{u}^{3} m \frac{d u}{d l} $$ (b) Under what condition does the classical equation \(F=m a\) hold? \(?\) (c) Assuming a constant force and that the speed is zero at \(t=0\), separate \(t\) and \(u\), then integrate to show that $$ u=\frac{1}{\sqrt{1+(F t / m c)^{2}}} \frac{F}{m} t $$ (d) Plot \(u\) versus \(t\). What happens to the velocity of an object when a constant force is applied for an indefinite length of time?

Short answer

Part (a) The relativistically correct expression for force is derived as \( F = \gamma_u^3 m du/dt \). Part (b) The classical force equation \( F = ma \) holds when the speed is much less than the speed of light and the motion is in a straight line. Part (c) The equation \( u = \sqrt(1 + (F t / m c)^2)^{-1} ( F t / m) \) represents the velocity as a function of time under a constant force. Part (d) From the graph of \( u \) against \( t \), we infer that the velocity of an object approaches the speed of light when a constant force is applied for an indefinite time.

Step-by-step solution

Part (a): Relativistically Correct Expression for Force

The relativistically correct expression for momentum, \(p\), is given by \( p = \gamma_u m u \) where \( \gamma_u = 1 / \sqrt{1 - u^{2} / c^{2}} \) where \( u \) is the velocity and \( c \) is the speed of light. Hence, \( F = dp/dt = d(\gamma_u m u)/dt \). With product rule of differentiation applied, \( F = \gamma_u^3 m du/dt \). This is the required expression.

Part (b): Condition for the Classical Equation F=ma

The classical equation \( F = m a \) holds under the following conditions: \n 1. The speed of the motion is much less than the speed of light i.e \( u << c \). This is because under such condition, we can approximate \( \gamma_u \approx 1 \). \n2. The force, mass and acceleration are in the same direction i.e the motion is in a straight line. When these conditions hold, the equation of motion simplifies to \( F = m a \).

Part (c): Derivation for the Velocity Equation

Assuming a constant force and that the speed is zero at \( t = 0 \), we can rewrite the equation derived in part (a) as \( du/dt = F/ (\gamma_u^3 m) \). With some manipulation of terms we can separate \( t \) and \( u \) as \( du / \sqrt(1-u^2/c^2)^3 = F dt / m \). \nAn integration on both sides give: \( \int_0^u du / \sqrt(1-(u/c)^2)^3 = \int_0^t F dt / m \). \nThe result of this integration will be \( u = \sqrt(1 + (F t / m c)^2)^{-1} ( F t / m) \).

Part (d): Plotting u(t) and Interpretation

The graph of \( u \) against \( t \) will have a hyperbolic shape. As \( t \) approaches infinity, the value of \( u \) approaches \( c \) i.e the speed of light. This shows that when a constant force is applied for an indefinite length of time, the velocity of an object approaches the speed of light. This aligns with the principle of relativity that nothing can travel faster than the speed of light.

Key concepts

Relativistic Momentum

In relativistic mechanics, the concept of momentum extends the classical idea to account for objects moving at speeds close to the speed of light. This is essential because, at high speeds, objects behave differently than what classical mechanics predicts.

• Relativistic momentum is defined as: \[ p = \gamma_u m u \] where \( p \) is momentum, \( m \) is mass, \( u \) is velocity, and \( \gamma_u \) is the Lorentz factor.
• The Lorentz factor \( \gamma_u \) is calculated as: \[ \gamma_u = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}} \] Here, \( c \) represents the speed of light.

The Lorentz factor becomes significant when the object's speed approaches the speed of light, making relativistic momentum larger than it would be if calculated classically. This formulation ensures that physical laws remain consistent across all inertial frames of reference.

Classical Mechanics

Classical mechanics provides the foundation for the motion of objects under the influence of forces when objects move at everyday speeds. It uses Newton's second law of motion, which states: \[ F = m a \] where \( F \) is force, \( m \) is mass, and \( a \) is acceleration.

• This equation is valid when the velocity of an object is much less than the speed of light (\( u << c \)). Under such conditions, the Lorentz factor \( \gamma_u \) approximates to 1, and relativistic effects can be ignored.
• The classical approach assumes that mass remains constant, and it applies well to motion along a straight line where force, mass, and acceleration align in the same direction.

When relativistic effects become negligible, predicting the motion of objects becomes much simpler using classical mechanics.

Speed of Light

The speed of light, denoted by \( c \), is a fundamental constant in physics. It represents the maximum speed at which all energy, matter, and information in the universe can travel. This constant plays a crucial role in both classical and relativistic physics:

• In classical mechanics, speeds much less than \( c \) simplify the equations of motion, allowing them to ignore relativistic effects.
• In relativistic mechanics, the speed of light signals the speed limit. As objects accelerate towards \( c \), their relativistic momentum grows, and their time experienced per unit of external time decreases.

This constant is instrumental in formulating theories like relativity, ensuring that no physical object can surpass this speed, aligning with Einstein's theories and leading to predictions that have been confirmed by a wealth of experimental data.

Equation of Motion

The equation of motion describes how objects move under various forces. In both classical and relativistic mechanics, this equation forms the backbone of analyzing dynamical systems:

• In classical mechanics, the familiar equation \( F = ma \) explains how an object's velocity changes under force, provided the speeds involved are far below \( c \).
• In relativistic mechanics, modifying this equation accounts for speeds nearing \( c \), as in the expression derived for force in relativistic scenarios: \[ F = \gamma_u^3 m \frac{du}{dt} \] This version accounts for changes in inertia due to increased velocity and ensures that all effects align with the relativistic framework.

The equation of motion drives the prediction and understanding of object trajectories, blending these fundamental prongs of physics whether describing a speeding car or particles in a collider at near-light speeds.