Question

A \(1000-\mathrm{kg}\) automobile is moving at \(20 \mathrm{~m} / \mathrm{s}\). Calculate the kinetic energy using both the nonrelativistic equation and the relativistic equation. What is the relative difference between these results?

Short answer

The non-relativistic kinetic energy of the car is approximately 200000J, which is the same as the relativistic kinetic energy. The relative difference between these two results is approximately 0%

Step-by-step solution

Calculate non-relativistic kinetic energy

For the non-relativistic kinetic energy, use the formula \(KE = \frac{1}{2}mv^2\). Substitute \(m = 1000kg\) and \(v = 20m/s\) to get \(KE_{non-rel} = \frac{1}{2} * 1000kg * (20m/s)^2 = 200000J\).

Calculate relativistic kinetic energy

Raise the velocity to the speed of light units using \(v = 20 m/s = 20 m/s * \frac{1}{3*10^8 m/s} = 6.7*10^-8c\). For the relativistic kinetic energy, use the formula \(KE = mc^2( \frac{1}{\sqrt{1−v^2/c^2}}−1)\). Here, \(m=1000kg, c=3*10^8 m/s\), and \(v=6.7*10^-8c\). Substitute these values to get \(KE_{rel} = 1000kg * (3*10^8 m/s)^2 ( \frac{1}{\sqrt{1−(6.7*10^-8)^2}}−1) \approx 200000J\)

Calculate the relative difference

To calculate the relative difference between these results, substituting in the formula will give \((\frac{KE_{non-rel}-KE_{rel}}{KE_{non-rel}})*100% = 0% \) due to both values being approximately similar.

Key concepts

Non-relativistic Kinetic Energy

When it comes to understanding kinetic energy in simpler terms, non-relativistic kinetic energy is often your go-to. It applies perfectly when dealing with objects moving much slower than the speed of light. The formula used here is \[ KE = \frac{1}{2}mv^2 \] In this equation:

• \( m \) represents the mass of the object, in kilograms (kg).
• \( v \) stands for the velocity, in meters per second (m/s).

For example, if a 1000-kg car travels at 20 m/s, you plug these values into the formula to find the kinetic energy. So, \( KE = \frac{1}{2} \times 1000 \times (20)^2 = 200,000 \text{ Joules} \). This calculation reveals the energy the car possesses due to its motion. Non-relativistic kinetic energy is straightforward and works when dealing with everyday speeds.

Relativistic Kinetic Energy

Unlike non-relativistic kinetic energy, relativistic kinetic energy must be considered when dealing with speeds that are a significant fraction of the speed of light. The formula is slightly more complex: \[ KE = mc^2 \left( \frac{1}{\sqrt{1-v^2/c^2}} - 1 \right) \] Here:

• \( m \) is the mass of the object.
• \( c \) symbolizes the speed of light, approximately \( 3 \times 10^8 \text{ m/s} \).
• \( v \) is the velocity of the object, expressed as a fraction of \( c \).

In the previous example, if the car's speed is converted to a fraction of light speed, \( v = 20 \text{ m/s} = 6.7 \times 10^{-8}c \), the relativistic kinetic energy is calculated using this formula. Despite the cumbersome math, the result remains around 200,000 Joules—similar to the non-relativistic case for such low speeds. Relativistic calculations become crucial only at very high speeds.

Relative Difference Calculation

When determining how much the non-relativistic and relativistic kinetic energies differ, you perform a relative difference calculation. This process tells us if the standard kinetic energy equation is sufficient for an object’s speed, or if relativistic effects must be factored in. The formula for relative difference is \[ \left( \frac{KE_{non-rel} - KE_{rel}}{KE_{non-rel}} \right) \times 100\% \] In the car example:

• Both kinetic energy values, non-relativistic and relativistic, were about 200,000 Joules.
• Plugging them into the relative difference formula gives 0\%.

This outcome confirms that relativistic effects are negligible at low speeds. Therefore, non-relativistic kinetic energy is sufficient for most everyday scenarios. Relative difference calculations are essential to identify situations where relativistic considerations become significant, such as when dealing with particles in a physics lab or satellites in space.