Question

An enemy spaceship is moving toward your starfighter with a speed, as measured in your frame, of 0.400c. The enemy ship fires a missile toward you at a speed of 0.700c relative to the enemy ship (Fig. E37.18). (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 * 106 km away from you when the missile is fired, how much time, measured in your frame, will it take the missile to reach you?

Short answer

(a) The missile's speed relative to you is approximately 0.859c. (b) It takes roughly 31.0 seconds for the missile to reach you.

Step-by-step solution

Understand the Scenario

You are in a starfighter and an enemy spaceship approaches you with a velocity of \( v_s = 0.400c \) (where \( c \) is the speed of light). The spaceship fires a missile towards you at \( v_m' = 0.700c \) relative to itself.

Use Relativistic Velocity Addition Formula

To find the speed of the missile concerning you, use the relativistic velocity addition formula: \[ v_m = \frac{v_m' + v_s}{1 + \frac{v_m' v_s}{c^2}} \] where \( v_s = 0.400c \) and \( v_m' = 0.700c \).

Substitute Values and Calculate Speed of the Missile

Substitute the given values into the formula: \[ v_m = \frac{0.700c + 0.400c}{1 + \frac{0.700c \times 0.400c}{c^2}} \]Simplify the equation: \[ v_m = \frac{1.100c}{1 + 0.280} = \frac{1.100c}{1.280} \]Calculating gives \( v_m \approx 0.859c \).

Calculate Time for Missile to Reach You

Given that the distance \( d = 8.00 \times 10^6 \) km, which is \( 8.00 \times 10^9 \) meters, use the equation \( t = \frac{d}{v_m} \).

Substitute Values and Solve for Time

Use the missile's speed concerning you: \( v_m = 0.859c \), so \[ t = \frac{8.00 \times 10^9}{0.859 \times 3.00 \times 10^8} \]Calculate \( t \approx 31.0 \) seconds.

Key concepts

Speed of Light

The speed of light, denoted by the symbol "c", is one of the most fundamental constants in physics. It holds a paramount position, especially in the realm of relativistic physics.
It is the maximum speed at which all massless particles and associated fields—including electromagnetic waves, such as light—travel in a vacuum.
- The value of the speed of light in a vacuum is approximately 299,792,458 meters per second (or roughly 300,000 kilometers per second). - In relativistic physics, "c" acts as an upper limit; nothing can travel faster than the speed of light.
In problems of relativistic kinematics, the speed of light is not just a measurable rate, but a guiding principle for predicting and understanding natural phenomena.
When calculating the velocity of objects moving at significant fractions of "c", it is crucial to apply the principles of relativistic kinematics rather than classical mechanics, as relativistic effects become non-negligible.

Relativistic Kinematics

Relativistic kinematics is a branch of physics that describes the motion of objects traveling at a significant fraction of the speed of light.
This area of study arises from the need to understand and predict the behavior of particles moving so fast that classical mechanics fails to provide accurate results.
- In these scenarios, Einstein's Special Theory of Relativity provides the framework for calculating and predicting the dynamics.- Relativistic velocity addition is an important concept here. It allows us to find the resultant velocity of objects moving close to the speed of light.
Standard addition of velocities, like you might have used in everyday problems, doesn't work at these high speeds due to effects predicted by relativity.
Instead, we use the relativistic velocity addition formula:\[ v = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}} \]This formula ensures that the resulting velocity never exceeds the speed of light, in line with relativistic principles. Using this formula in the problem at hand allowed us to determine that the missile's speed relative to the observer is approximately 0.859c.

Time Dilation

Time dilation is a fascinating consequence of Einstein's theory of relativity. In simple terms, it describes how time elapses at different rates for observers depending on their relative motion and gravitational field.
- When dealing with objects moving at relativistic speeds, time doesn't pass at the same rate for all observers. Instead, it slows down for the moving object relative to a stationary observer. - In the context of fast-moving objects, like the starfighter and missile in our exercise, time dilation is considered when calculating travel and reaction time.
Although our primary solution deals with velocities towards a moving observer, the theory provides insight into how different inertial frames of reference can lead to different measurements of time intervals.
This aspect of special relativity not only applies when determining the time for an event—like the missile reaching you—but also when considering the relative perceptions of time for the spacefaring enemies and the one in the starfighter.