3.6. Numerical Problems

Today, we'll start by learning about how to find the resultant velocity when two velocities act at right angles to each other. Can anyone tell me what formula we might use?

Is it the Pythagorean theorem?

Exactly! We use $v_R = \sqrt{v_1^2 + v_2^2}$ where $v_R$ is the resultant velocity. Let's use an example: a bird flying at 50 m/s east and 60 m/s north. What would the resultant velocity be?

I think it's $\sqrt{(50)^2 + (60)^2}$ which equals 78.10 m/s.

Perfect! And now, how about the direction?

We would use $\theta = \tan^{-1}(60/50)$, so it's approximately 50.19 degrees north of east.

Great work, everyone! Let's summarize the key points: we find the resultant velocity using the Pythagorean theorem and the direction using tangent.

Now, let's shift gears and talk about resolving velocity. Who can explain what we mean by 'resolving' a velocity vector?

It's breaking down a velocity into its components, usually horizontal and vertical.

Exactly! For example, if a plane flies at 200 m/s at an angle of 40 degrees to the horizontal, how do we find its components?

We can use trigonometric functions: $v_x = v \cos(\theta)$ and $v_y = v \sin(\theta)$.

That's right! So for our plane, what do we get for each component?

The horizontal component $v_x$ is about 153.2 m/s and $v_y$ is about 128.6 m/s.

Fantastic! Remember, this resolution helps us analyze the motion of objects more easily. Let's recap: we resolve velocity using cosine and sine based on the angle given.