Answer :
Answer:
Sure, I can guide you through solving problems related to calculating trajectories of space missions involving distance, speed, and time. Let's go through an example step-by-step:
**Problem:**
A spacecraft is launched from Earth with an initial speed of 25,000 meters per second. Calculate the following:
(a) How far will the spacecraft travel in 10 hours?
(b) How much time will it take for the spacecraft to reach a distance of 1,000,000 kilometers from Earth?
**Given:**
Initial speed of spacecraft, \( v_0 = 25,000 \) m/s
**Solution:**
**(a) Distance traveled in 10 hours:**
To find the distance traveled, we use the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
Given:
- Initial speed, \( v_0 = 25,000 \) m/s
- Time, \( t = 10 \) hours
First, convert hours to seconds (since speed is in meters per second):
\[ t = 10 \times 3600 \text{ seconds} = 36,000 \text{ seconds} \]
Now, calculate the distance:
\[ \text{Distance} = v_0 \times t \]
\[ \text{Distance} = 25,000 \text{ m/s} \times 36,000 \text{ s} \]
\[ \text{Distance} = 900,000,000 \text{ meters} \]
So, the spacecraft will travel **900,000,000 meters** in 10 hours.
**(b) Time taken to reach 1,000,000 kilometers:**
To find the time taken, rearrange the distance formula:
\[ t = \frac{\text{Distance}}{\text{Speed}} \]
Given:
- Distance, \( \text{Distance} = 1,000,000 \) kilometers = \( 1,000,000 \times 1000 \) meters (since 1 km = 1000 m)
\[ \text{Distance} = 1,000,000,000 \text{ meters} \]
Now, calculate the time:
\[ t = \frac{1,000,000,000 \text{ meters}}{25,000 \text{ m/s}} \]
\[ t = 40,000 \text{ seconds} \]
Convert seconds to hours:
\[ t = \frac{40,000}{3600} \text{ hours} \]
\[ t \approx 11.11 \text{ hours} \]
Therefore, it will take approximately **11.11 hours** for the spacecraft to reach a distance of 1,000,000 kilometers from Earth.
These calculations involve basic principles of distance, speed, and time, which are fundamental in understanding trajectories of space missions. If you have more specific scenarios or further questions, feel free to ask!