Question

Consider a particle moving with a speed less than \(0.5 c .\) If the speed of the particle is doubled, by what factor will the momentum increase? a) less than 2 b) 2 c) greater than 2

Short answer

Answer: greater than 2

Step-by-step solution

Write the initial momentum formula

Use the relativistic momentum formula to represent the initial momentum where \(p_1 = \frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}\).

Write the momentum formula with doubled speed

Use the same relativistic momentum formula with doubled speed (2v) to represent the momentum where \(p_2 = \frac{m(2v)}{\sqrt{1-\frac{(2v)^2}{c^2}}}\).

Find the ratio of the final and initial momentum

Divide the final momentum formula by the initial momentum formula to find the ratio or the factor by which the momentum increases: \(\frac{p_2}{p_1} = \frac{\frac{m(2v)}{\sqrt{1-\frac{(2v)^2}{c^2}}}}{\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}}\)

Simplify the equation

After simplification, we obtain: \(\frac{p_2}{p_1} = \frac{2}{\sqrt{\frac{1-\frac{v^2}{c^2}}{1-\frac{4v^2}{c^2}}}}\)

Compare the ratio to 2

We need to compare the ratio \(\frac{p_2}{p_1}\) to 2 in order to decide which option (a, b, or c) is correct. Since the initial speed \(v\) is less than 0.5c, \(\frac{v^2}{c^2}\) is less than 0.25, and \(\frac{4v^2}{c^2}\) is less than 1. Therefore, the ratio \(\frac{p_2}{p_1}\) is greater than 2. Thus, the correct answer is option c) greater than 2.

Key concepts

Special Relativity

Special relativity is a groundbreaking theory of physics formulated by Albert Einstein in 1905. It revolutionized our understanding of time, space, and energy. The theory includes several intriguing and non-intuitive concepts, such as the idea that the laws of physics are the same for all observers in uniform motion relative to one another, no matter their velocity.

One of the key postulates of special relativity is that the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or observer. This has incredible implications, such as time dilation and length contraction, which mean that time can pass at different rates and objects can appear shorter depending on the relative velocity between observers.

These effects become significant at speeds approaching the speed of light, denoted by the symbol 'c'. It's important to distinguish that these predictions are not noticeable in everyday life, as our common experiences occur at speeds much slower than light speed.

Momentum Increase Factor

In the realm of classical physics, momentum is simply the product of an object’s mass and its velocity. However, when dealing with velocities that are a significant fraction of the speed of light, such as in the aforementioned exercise, the classical definition falls short. The momentum increase factor comes into play as a way to account for the effects of special relativity on moving objects.

Instead of being a constant multiple, the momentum of an object increases more than its speed when the speeds are relativistic. This is because the momentum of an object in relativity is given by the formula \( p = \frac{mv}{\sqrt{1-\frac{v^2}{c^2}}} \), where 'm' is the rest mass, 'v' is its velocity, and 'c' is the speed of light. The denominator includes a relativistic factor that accounts for the increased momentum, and this factor grows smaller as the velocity approaches the speed of light, leading to the momentum increasing by a greater factor than the velocity.

Therefore, when velocity is doubled in relativistic scenarios, we don't simply double the momentum. This is why the solution shows that when the velocity of an object is doubled, its momentum increases by a factor greater than 2.

Velocity in Special Relativity

When we talk about velocity in special relativity, we're discussing much more than just speed. We consider the implications of moving close to the speed of light and how these movements affect measurements of time and space. Unlike in classical mechanics, velocities don't simply add together in special relativity because of the effects imposed by the constancy of the speed of light.

The consequence of this is that velocities are not directly proportional to momenta at high speeds. As an object’s velocity increases and gets closer to 'c', the less its velocity can increase with the same force applied. Also, the relativistic formula for momentum indicates there is a significant increase for velocity changes at high speeds. The work done on a particle moving at relativistic speeds does much more than increase its velocity; it increases its mass-energy, which in turn significantly increases its momentum.

This nuance is critical in understanding problems like the one in the exercise, where doubling the speed does not double the momentum, as it would in classical, non-relativistic scenarios. A detailed analysis of the increase in momentum in comparison to the increase in speed reveals the extraordinary effect of relativity on high-speed particles.