![]() It's usually simpler to adjust the angle of the barrel than to adjust the bullet speed (although that may have been less the case for cannon in Galileo's day). For a gun, of course, the more natural variables would be the initial speed of the bullet v and the angle q of the gun barrel to the horizontal. We've so far followed Galileo's analysis, treating the horizontal and vertical motions separately. ![]() The reason to keep resaving the spreadsheet under new names is that if it gets too messed up, you can always go back a few steps, not to the beginning! This translates as "If the last two horizontal positions of the particle in the table are the same-so it's come to rest-then range = the horizontal position, otherwise write "still in air". In A11 we write range=, and in B11 we write: ![]() ![]() Of course, we could figure this out by looking at the graph, but it's nice to have it done automatically. This would be the maximum value of x it attains, except that if we choose delta_t too small, it might still be in the air at the end of the 200 rows of calculation, so then the value of x is not the true range. To get a bit more practice with IF statements, let us find the range of the projectile, how far away it lands. We then copy these formulas down to D214, E214. This stops the ball falling further, but if we want it to really stay where it is we must also stop the horizontal motion! So, in D16 we write: =IF(E15+0.5*(C15+C16)*delta_t>0,D15+0.5*(B15+B16)*delta_t,D15) BUT if this step is going to get you below ground level, don't do it-stay where you are, just put E16 = E15. This means that as long as the ball will still be above ground after this step, do what you were doing. We want to tell the spreadsheet that if it finds the ball will be underground on the next step, stop right there! (Of course, this means we'll stop the ball slightly above ground level, but if the step size is small, this won't be a big error, and we'll ignore it for now.)Įxcel has an IF function. Assuming we're throwing a ball in a level field, this is an undesirable feature-we'd like it to stop when it gets to ground level. One problem with this spreadsheet as it stands is that it doesn't know when to stop-the ball falls back to ground level, then continues right on into the ground. We're going to do some more work on it, but don't want to lose what we've done so far. Having done that, save it again as Projectile2. NOW SAVE THIS SPREADSHEET AS PROJECTILE1. Call the graph Projectile: Zero Air Resistance, label the axes " distance along ground" and " height above ground". Highlight cells D15 through E214, and click Chartwizard. Now, select cells A16 through E16 and copy all five columns down through E214. (Also, Bold and Right Justify A7, A8, A9, A10.) Then select these cells, click Bold, and Center justify. Put the appropriate names in B8, B9 and B10, and then enter some reasonable values, say, 10, 20, 30, 0.05, ready for when we construct the table. Click on B7, click Insert/Name/Define, it will suggest name g, click OK. In A7, A8, A9 and A10 write respectively g=, v_x_init=, v_y_init= and delta_t=. (Of course, v_x isn't going to change, but we're going to need that column when we include air resistance, so we might as well put it in now.) Since we're interested in both velocity and position of the projectile as functions of time, we'll construct a spreadsheet with five columns: time, v_x, v_y, x, y. Of course, we also need to specify the time interval used in our discretization of the motion, we'll call it delta_t as usual. In contrast to our earlier spreadsheets on falling objects, we will now take the upward direction to be positive. There are three variables: the initial horizontal velocity, call it v_x_init, the initial vertical velocity v_y_init, and the acceleration due to gravity g. Let's call it Projectile1, and write in A1 " Motion of a Projectile Under Gravity" It's easy to reproduce this compound motion with a spreadsheet. Galileo argues that, if air resistance could be neglected, the horizontal motion was one at constant velocity, the vertical motion was one of uniform downward acceleration, identical to that of an object falling straight down. The first successful attempt to describe projectile motion quantitatively followed from Galileo's insight that the horizontal and vertical motions should be considered separately, then the projectile motion could be described by putting these together. HOME Using Spreadsheets for Projectile Motion
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