Relativistic Baseball. Calculate the magnitude of the force required to give a 0.145 kg baseball an acceleration a = 1.00 m/s2 in the direction of the baseball's initial velocity when this velocity has a magnitude of (a) 10.0 m/s; (c) 0.990c.

Ch 37: Special Relativity

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 37: Special Relativity

Problem 26ac

Chapter 37, Problem 26ac

Relativistic Baseball. Calculate the magnitude of the force required to give a 0.145 kg baseball an acceleration a = 1.00 m/s² in the direction of the baseball's initial velocity when this velocity has a magnitude of (a) 10.0 m/s; (c) 0.990c.

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Step 1: Begin by recalling Newton's second law of motion, which states that the force is equal to the rate of change of momentum: F = dp/dt. For relativistic cases, momentum is given by p = γmv, where γ = 1 / √(1 - v²/c²) is the Lorentz factor, m is the mass, and v is the velocity.

Step 2: For each velocity scenario, calculate the Lorentz factor γ using the formula γ = 1 / √(1 - v²/c²). Substitute the given velocities (10.0 m/s, 0.900c, and 0.990c) into the equation, where c is the speed of light (approximately 3.00 × 10⁸ m/s).

Step 3: Determine the relativistic mass using m_rel = γm, where m is the rest mass of the baseball (0.145 kg). This accounts for the increase in mass due to relativistic speeds.

Step 4: Use the relativistic form of Newton's second law, F = γma, to calculate the force for each scenario. Substitute the values of γ, m, and the given acceleration a = 1.00 m/s² into the equation.

Step 5: Ensure that the units are consistent throughout the calculations (e.g., kilograms for mass, meters per second for velocity, and meters per second squared for acceleration). The resulting force values will be in newtons (N).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Newton's Second Law of Motion

Newton's Second Law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). This fundamental principle allows us to calculate the force required to accelerate an object, such as a baseball, under various conditions. Understanding this law is crucial for solving problems involving motion and force.

Relativistic Effects

At speeds approaching the speed of light, relativistic effects become significant, altering the mass and behavior of objects. The relativistic mass increases as an object's velocity approaches the speed of light (c), which affects the force required to achieve a given acceleration. This concept is essential for understanding how forces behave at high velocities, such as 0.900c and 0.990c.

Acceleration

Acceleration is defined as the rate of change of velocity of an object over time. It is a vector quantity, meaning it has both magnitude and direction. In this problem, the baseball is subjected to a specific acceleration of 1.00 m/s², which is crucial for determining the force needed to achieve this change in velocity, especially when considering the effects of relativistic speeds.

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