Vectors and Beyond (two problems)

[pippintook]

First Problem- Nursery Rhyme Style:

When the three blind mice were running after the farmer's wife, they first ran two meters east, then three meters north, and then four meters southwest, at which point the farmer's wife lay hold to a carving knife. How far are the mouse-tails from their original point of departure?

After drawing it out and trying various equations, I am pretty sure it's close to two meters. But I don't know how to solve it.Second Problem- Astronaut Style

A spacecraft is maneuvering in Cartesian Space at 30 m/s to fetch some space debris as part of Operation Clean Sweep. The debris is floating along the x-axis at a constant speed of 10 m/s. If the debris when first sighted is at coordinates (100, 0), and if the spacecraft is at coordinates (100, 100), in what direction should the spacecraft proceed in order to intercept the debris?

This one...I drew it out, but again, have no idea what to do.

Help please?

 

[Kurdt]

The first question is basically vector addition. Have you been taught that?

For the second part you need to set up some equations of motion. You know that the spacecraft and the debris must get to the same point on the x-axis at the same time if they are to intercept.

 

[baba89]

I would approach these problems by breaking them down into smaller, more manageable steps. For the first problem, we can use basic vector addition to find the displacement of the mice from their original starting point. We know that they first ran 2 meters east, then 3 meters north, and finally 4 meters southwest. Using a coordinate system, we can see that the mice ended up 1 meter south and 1 meter west from their starting point. Therefore, the total displacement is √(1^2 + 1^2) = √2 meters.

For the second problem, we can use the concept of relative velocity to determine the direction in which the spacecraft should proceed to intercept the debris. We know that the debris is moving at a constant speed of 10 m/s along the x-axis, and the spacecraft is moving at 30 m/s at a 45 degree angle from the x-axis. We can use trigonometry to find the component of the spacecraft's velocity that is parallel to the debris's velocity, which is 30 m/s * cos(45) = 21.2 m/s. This means that the spacecraft needs to decrease its x-component velocity by 21.2 m/s in order to match the debris's velocity and intercept it. Therefore, the spacecraft should proceed in a direction of 45 degrees counterclockwise from the x-axis.

I hope this helps and provides a scientific approach to solving these problems. Remember, breaking down complex problems into smaller steps and utilizing relevant concepts and equations can help make them more manageable!