Problem 22 - University Physics with Modern Physics

Question

An enemy spaceship is moving toward your starfighter with a speed, as measured in your frame, of 0.400 \(\mathrm{c}\) . The enemy ship fires a missile toward you at a speed of 0.700\(c\) relative to the enemy ship (Fig. 37.28\()\) . (a) What is the speed of the missile relative to you? Express your answer in terms of the speed of light. (b) If you measure that the enemy ship is \(8.00 \times 10^{6} \mathrm{km}\) away from you when the missile is fired, how much time, measured in your frame, will it take the missile to reach you?

Step-by-Step Solution

Answer

(a) 0.859c. (b) 31.1 seconds.

Step 1: Understand the Problem

We need to find the speed of the missile relative to the starfighter, taking into account relativistic speed addition since velocities are a significant fraction of the speed of light, \(c\). The scenario includes an enemy spaceship moving towards the starfighter at 0.400\(c\) and a missile moving relative to the enemy spaceship at 0.700\(c\). Also, we need to calculate how long it takes for the missile to reach the starfighter if the spaceship is initially 8.00 \times 10^6 km away.

Step 2: Use the Relativistic Velocity Addition Formula

The formula for relativistic velocity addition is given by:\[v = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}}\]where \(v\) is the velocity of the missile relative to the starfighter, \(v_1\) is the velocity of the spaceship relative to the starfighter (0.400\(c\)), and \(v_2\) is the velocity of the missile relative to the spaceship (0.700\(c\)). Substitute the values into the formula:\[v = \frac{0.400c + 0.700c}{1 + \frac{(0.400)(0.700)}{1}} = \frac{1.100c}{1.280} \approx 0.859c\].

Step 3: Compute Distance and Time Relationship

Given that the spaceship is 8.00 \times 10^6 km away and the missile moves at a speed of 0.859\(c\) relative to the starfighter, we use the formula:\[\text{time} = \frac{\text{distance}}{\text{speed}}\]Convert the distance from km to m: 8.00 \times 10^6 km = 8.00 \times 10^9 m. Calculate time:\[time = \frac{8.00 \times 10^9 \, \text{m}}{0.859c} = \frac{8.00 \times 10^9 \, \text{m}}{0.859 \times 3.00 \times 10^8 \, \text{m/s}} \approx 31.1 \, \text{s}\].

Key Concepts

Special Relativity

Einstein's theory of Special Relativity revolutionized our understanding of time and space. It introduced groundbreaking concepts such as time dilation, length contraction, and the relativity of simultaneity. A critical aspect of this theory is that the laws of physics are the same in all inertial frames, and the speed of light is constant for all observers, regardless of their motion. Relativistic effects become significant when objects move at speeds close to the speed of light, denoted by the symbol \( c \).

This theory challenges classical mechanics, where velocities simply add up in a straight-forward manner. Instead, Special Relativity requires us to apply a relativistic formula, especially when dealing with problems involving high speeds. As in the exercise with the starfighter and spaceship, these effects must be considered when calculating the speed of objects like missiles relative to each other.

Relativistic Physics

Relativistic Physics is the study of how physical processes operate in a regime where the speed of light cannot be considered infinitely fast compared to the motion of particles. One key component is understanding how velocities add up when an object is moving in a frame that is itself moving. This is necessary for calculating the speed of the missile relative to the starfighter.

In non-relativistic physics, an object moving at 30 m/s on a train traveling at 50 m/s would simply be added together to find a speed of 80 m/s. However, at relativistic speeds significant fractions of the speed of light, this addition does not hold true. Instead, the relativistic velocity addition formula is used:

\[v = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}}\]

Here, \(v_1\) and \(v_2\) are the object's velocities in different reference frames. The formula accounts for the fact that as velocities approach the speed of light, they don't simply add, due to the nature of space and time as described by Special Relativity.

Velocity Calculations

Calculating velocities in a relativistic context involves using the relativistic velocity addition formula to accurately account for high-speed movements. In our problem, the enemy spaceship is moving at 0.400\(c\) toward the starfighter, and it fires a missile at 0.700\(c\) relative to itself.

Applying the relativistic velocity addition formula, we find the speed of the missile relative to the starfighter. Substituting these values, we calculate:

\[v = \frac{0.400c + 0.700c}{1 + \frac{0.400 \times 0.700}{1}} = \frac{1.100c}{1.280} \approx 0.859c\]

Once we have this velocity, the next step is determining how long it takes for the missile to reach the starfighter. With the enemy spaceship 8.00 \(\times 10^6\) km away, we find the time using:

\[\text{time} = \frac{\text{distance}}{\text{speed}}\] Convert the distance to meters and use the speed of light \(c = 3.00 \times 10^8\) m/s to calculate the time:

\[\text{time} = \frac{8.00 \times 10^9 \, \text{m}}{0.859 \times 3.00 \times 10^8 \, \text{m/s}} \approx 31.1 \, \text{s}\]

This procedure highlights the need for precision and understanding when working with high-speed velocities in the realm of physics.