- #1

opus

Gold Member

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## Main Question or Discussion Point

Let me start off by stating a given problem:

A baseball is hit, and leaves the bat at a speed of 100 mph and at an angle of 20° from the horizontal. Express this velocity in vector form.

So we're given the velocity and the angle at which the ball is hit. The speed corresponds to the vector's magnitude, and the angle corresponds to the vector's direction. To get this into vector form we can use the fact that:

$$u=\left<\left\|u\right\|cos(θ), \left\|u\right\|sin(θ)\right>$$

$$=\left\|u\right\|\left<cos(θ),sin(θ)\right>$$

So in this case,

$$=\left\|u\right\|\left<cos(θ),sin(θ)\right>$$

$$=100\left<cos(20°),sin(20°)\right>$$

$$≈\left<93.97,34.20\right>$$

Now here's my question:

These components don't seem to tell nearly as much as the initial problem stated. In looking at the components, you can't tell the velocity of the ball, or the angle at which it was hit. We can just see the horizontal and vertical displacement. So what's the point of getting anything into vector form like this?

A baseball is hit, and leaves the bat at a speed of 100 mph and at an angle of 20° from the horizontal. Express this velocity in vector form.

So we're given the velocity and the angle at which the ball is hit. The speed corresponds to the vector's magnitude, and the angle corresponds to the vector's direction. To get this into vector form we can use the fact that:

$$u=\left<\left\|u\right\|cos(θ), \left\|u\right\|sin(θ)\right>$$

$$=\left\|u\right\|\left<cos(θ),sin(θ)\right>$$

So in this case,

$$=\left\|u\right\|\left<cos(θ),sin(θ)\right>$$

$$=100\left<cos(20°),sin(20°)\right>$$

$$≈\left<93.97,34.20\right>$$

Now here's my question:

These components don't seem to tell nearly as much as the initial problem stated. In looking at the components, you can't tell the velocity of the ball, or the angle at which it was hit. We can just see the horizontal and vertical displacement. So what's the point of getting anything into vector form like this?