Question

During a one hour trip, a small boat travels 80.0 km North and then travels 60.0 kn East.1. What is the boats displacement for the one hour trip?2. What is the boats average speed durin the one hour trip?3. What is the magnitude of the boats average velocity for the one hour trip?4. What is the direction of the boats average velocity for the one hour trip?

Answer 1

Let us first draw, in order to better understand the problem:

1) The boat's displacement

To calculate this is to find the hipothenuse on the triangle above. This can be achieved through Pythagoras theorem:

`c^2=80^2+60^2\rightarrow c=\sqrt[\placeholder{⬚}]{80^2+60^2}=100km`

So, the total displacement is 100km

2) The boat's average speed

Considering the boat traveled 140km in one hour, we can calculate its average speed through the formula:

`v=(\Delta s)/(\Delta t)=(140km)/(1h)=140(km)/(h)`

Thus, the average speed is 140km/h

3) The magnitude of the average velocity

When we talk about velocity, it concerns not only the displacement, but also the direction in which it is going. The average velocity is given by the magnitude of the vectorial difference between the start point and end point divided by the time it took to travel. With this we'll have:

`|\vec{v}|=(100km)/(1h)=100(km)/(h)`

4) The direction of the boat's average velocity

First, let us draw the situation as vectors:

From this drawing, we can see that the vectors for displacement and velocity are parallel, thus, the triangle formed by the displacement vectors is similar to the one formed by the velocity vectors. We can then find the following angle:

By using the information from the displacement triangle, we get:

`sin(\theta)=(80)/(100)=0.8\rightarrow\theta=arcsin(0.8)\approx53.13°`

Note:

On velocity and speed, this drawing may make it more clear: