Mechanics: Chapter 2 – Motion in Two or More Dimensions

Motion in more than one dimension relies on the same principles as motion in one dimension. The trick is two create a convenient co-ordinate system and break the motion down in separate components. The most common example used to highlight this is projectile motion. Neglecting air resistance and assuming the gravitational acceleration remains constant etc. a projectile has essentially a vertical gravitational acceleration acting on it and no horizontal acceleration. In other words, a_{x} = 0 while a_{y} = -g (assuming we’re taking the downwards direction as negative in our co-ordinate system. Since the accelerations are constant, we can just substitute these values into the equations of motion to get

  • v_{x} = v_{0x}
  • v_{y} = v_{0y} -gt
  • x = v_{0x}t
  • y = v_{0y}t - \frac{1}{2}gt^2

If we’re lazy enough to chose a co-ordinate system with the x and y-axes perpendicular to one another, we can use elementary trigonometry to separate the components of velocity. In other words,

  • v_{x} = vcos(\theta)
  • v_{y} = vsin(\theta)

where \theta is the angle of the velocity vector from the horizontal.

We can also break stuff up into fancy unit vectors like this:

  • \hat{v} = v_{x}\hat{i} + v_{y}\hat{j} = |\hat{v}|cos(\theta)\hat{i} + |\hat{v}|sin(\theta)\hat{j}
  • Similarly, for a position vector r: \hat{r} = x\hat{i} + y\hat{j} = |\hat{r}|cos(\theta)\hat{i} + |\hat{r}|sin(\theta)\hat{j}

In fact, breaking things up into unit vectors is a pretty sleek way of resolving motion in any number of dimensions. Just remember to choose convenient co-ordinate systems and separate out the components of vectors correctly.

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