![]() Imagine if you had to rederive the Pythagorean theorem every time you wanted to use it instead of just being able to plug the numbers into the formula. Useful for the military, projectile motion can now be used for a number of weapons which is when an object (like a bullet or cannon) is thrown-projected- and. Also, once you have a general expression for a thing, you've essentially solved that class of problem. In general, whenever you can – that is, whenever it's not prohibitively difficult – you should try to solve the thing symbolically to gain the greatest insight. For example, Maybe the expression for the area of a circle shows up somewhere in the final expression, which can suggest a different derivation or interpretation. Maths version of what Teacher Mackenzie said: Find the time it takes for an object to fall from the given height. The projectile still falls below its gravity-free path by a vertical distance of 0.5gt2. These distances are indicated on the diagram below. The projectile still falls 4.9 m, 19.6 m, 44.1 m, and 78.4 m below the straight-line, gravity-free path. ![]() But when you solve the thing symbolically, you can interpret the equation, see clearly what's proportional to what, any algebraic symmetry (functional symmetry, being able to swap variables, so on), you can see patterns or that some other quantity might be hidden in the thing. The diagram below depicts the position of a projectile launched at an angle to the horizontal. ![]() When you solve a thing numerically, you just get some number (or a vector, etc.) at the end (and maybe some units). ![]() Yeah, and it's actually a great way to gain insight into the nature of the thing. 8 s m ) 2 (plug in horizontal and vertical components of the final velocity) v, squared, equals, left parenthesis, 7, point, 00, start fraction, start text, m, end text, divided by, start text, s, end text, end fraction, right parenthesis, squared, plus, left parenthesis, minus, 20, point, 8, start fraction, start text, m, end text, divided by, start text, s, end text, end fraction, right parenthesis, squared, start text, left parenthesis, p, l, u, g, space, i, n, space, h, o, r, i, z, o, n, t, a, l, space, a, n, d, space, v, e, r, t, i, c, a, l, space, c, o, m, p, o, n, e, n, t, s, space, o, f, space, t, h, e, space, f, i, n, a, l, space, v, e, l, o, c, i, t, y, right parenthesis, end text ![]()
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