# Calculate the speed of the spacecraft

• kreil
In summary, the question involves finding the speed of a spacecraft as measured by an Earth observer. One approach is to use the formula for length contraction and solve for the velocity. Another approach is to consider the interval between two events and use the invariance of the interval to find the time difference from the spacecraft's point of view.
kreil
Gold Member

## Homework Statement

A spacecraft with a proper length of 300m takes 0.75E-6s to pass an Earth observer. Calculate the speed of the spacecraft as measured by the Earth observer.

## The Attempt at a Solution

I'm not sure how to get this, when you plug in those numbers you get 4E8m/s, which is faster than the speed of light. This makes me think that one of the given data needs to be adjusted, but length/time changes all include u, which is what I am trying to solve, and this leads to 2 unknowns in 1 equation. If anyone can give me a hint I would appreciate it.

Can you show exactly what you've tried so far? It'll allow us to help you better.

Remember that the length of the craft with respect to the Earth is not 300m, but shorter due to Lorentz contraction.

well $$v=\frac{d}{t}=\frac{300m}{75 microseconds}=4x10^8m/s$$

that is what i was talking about before.

so if the length is adjusted how do i use $$L=L_0\sqrt{1-u^2/c^2}$$ here, since I'm trying to find u and L in the same equation?

I think this would work: Consider the two events, the nose of the craft passing you, the Earth observer, and the tail of the craft passing you. Find the value of the interval $$\Delta s^2$$. Consider the same events from the space-traveller's point of view. You know the interval is invariant. So find the time between the events from the traveller's point of view, and use it to find the relative velocity.

I'm not sure if it's the most efficient method, but it is worth a shot.

kreil said:
well $$v=\frac{d}{t}=\frac{300m}{75 microseconds}=4x10^8m/s$$

that is what i was talking about before.

so if the length is adjusted how do i use $$L=L_0\sqrt{1-u^2/c^2}$$ here, since I'm trying to find u and L in the same equation?

300m is the PROPER length of the craft. If you stand still and the craft passes by you, would you use the proper length or the length as measured by you? (Hint: you want the speed of the craft as measured by the Earth observer. It's done by recording the position and the time of the nose and the time when the tail touches that position, then dividing the length of the craft by the time difference).

kreil said:
well $$v=\frac{d}{t}=\frac{300m}{75 microseconds}=4x10^8m/s$$

that is what i was talking about before.

so if the length is adjusted how do i use $$L=L_0\sqrt{1-u^2/c^2}$$ here, since I'm trying to find u and L in the same equation?

No, you are not trying to find L- you are not asked for that. Just use that formula for L rather than 300m in your original formula:
$$u= \frac{300m\sqrt{1-u^2/c^2}}{75\cdot 10^{-6}}$$
and solve the equation for u.

thanks for the help!

## 1. How do you calculate the speed of a spacecraft?

The speed of a spacecraft can be calculated by dividing the distance traveled by the time it took to travel that distance. This is known as the speed formula: speed = distance/time.

## 2. What units are used to measure the speed of a spacecraft?

The speed of a spacecraft is typically measured in kilometers per hour (km/h) or meters per second (m/s). Other common units include miles per hour (mph) and feet per second (ft/s).

## 3. Can the speed of a spacecraft change?

Yes, the speed of a spacecraft can change. This can happen due to external forces such as gravity, friction, or thrust from engines. It can also change due to internal factors such as fuel consumption or engine performance.

## 4. What is the difference between average speed and instantaneous speed?

Average speed is the total distance traveled divided by the total time taken, while instantaneous speed is the speed at a specific moment in time. Average speed gives an overall picture of how fast the spacecraft is moving, while instantaneous speed gives a more precise measurement at a particular point.

## 5. How accurate are speed calculations for spacecraft?

Speed calculations for spacecraft can be very accurate when all factors are taken into account. However, small variations in factors such as atmospheric conditions or gravitational pull can affect the speed, so it is important to continuously monitor and adjust calculations for accuracy.

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