Using Spreadsheets for Projectile Motion

8/12/2019 Using Spreadsheets for Projectile Motion

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Using Spreadsheets for Projectile Motion

Michael Fowler

University of Virginia

Putting Galileo's Ideas in a Spreadsheet

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.

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.

It's easy to reproduce this compound motion with a spreadsheet. et's call it Projectile1, and

write in !"#Motion of a Projectile Under Gravity#

There are three variables$ the initial horizontal velocity, call it v%&%init, the initial vertical

velocity v%y%init, and the acceleration due to gravity g.In contrast to our earlier

spreadsheets on falling objects, we will now take the upwarddirection to be positive.

f course, we also need to specify the time interval used in our discretization of the motion,

we'll call it delta%t as usual.

(ince 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%&, v%y, &, y. )f course, v%& 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.*

In !+,!,!-and !"write respectively g=, v_x_init=, v_y_init=and delta_t=. /lic0 on

1+, clic0 Insert23ame24efine, it will suggest name g, clic0 5. 6ut the appropriate names in

1, 1-and 1", and then enter some reasonable values, say, 10, 20, 30, 0.05, ready for when

we construct the table.

3ow, in !"7, 1"7, /"7, 4"7and 8"7write time, v_x, v_y, x, y. Then select these cells,

clic0 1old, and /enter justify. )!lso, 1old and 9ight :ustify !+, !, !-, !".*

In !";, 1";, /";, 4";and 8";write 0, v_x_init, v_y_init, 0and 0.

In !"


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A4( 67 P+(A$6(( A P+8("79(1.

>aving done that, save it againas Projectile2. ?e're going to do some more wor0 on it, but

don't want to lose what we've done so far.

Stop the Ball When It Hits the Ground!

ne problem with this spreadsheet as it stands is that it doesn't 0now when to stop@the ball

falls bac0 to ground level, then continues right on into the ground. !ssuming we're throwing

a ball in a level field, this is an undesirable feature@we'd li0e it to stop when it gets to ground

level.

?e want to tell the spreadsheet that if it finds the ball will be underground on the ne&t step,

stop right thereA )f 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.*

Using the IF Function

8&cel has an IB function. It's written IB)(tatement, C,D*. #(tatement# is some logical

statement, such as 8"EF. If the statement is true, 8&cel does C. If it's false, 8&cel does D.

3ow, in 8"


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Air esistance

!s usual, resavethe spreadsheet as 6rojectile7. ?e are about to add air resistance.

et's assume that the air resistance is proportional to the square of the velocity, and, of course

is directed bac0wards, so it has magnitude @bv=, where bis a coefficient and vis the speed ofthe projectile.

It's useful to 0eep trac0 of the speed of the projectile, so we'll put in an e&tra column for that.

/lic0 on the 4 above the cell 4". This will select the whole column which trac0s the x@

position. 3ow clic0 Insert2/olumns and a new blan0 column will appear. The new column

will be 4, thex@positions are now in 8. )8&cel will automatically adjust formulas.* ?rite #v#

in 4"7.

8nter in the new 4";$ =C+%!15#!15"15#"15'

/opythis formula down to the end of the table.

?e're now ready to include the drag force in the equations of motion. These, of course, give

the rate of change of the horizontal and vertical components of the velocity, in other words

the vector equation FH mais split into componentsF&H ma&,FyH may. p to this point, we

have hadF&H andFyH @mg. ?e must now add the appropriate components of the drag

force. It is a vector of magnitude bv=, and direction antiparallel to v. >ence itsx@ andy@

components are in thesame ratioto its total length as the corresponding components of v. (o,

the components of drag force felt by the projectile are @)v&2v*bv=and @)vy2v*bv

=.

The full equations of motion are$

ma&H @ )v&2v*bv=, mayH @mg@ )vy2v*bv

=, )we'll ta0e mH " for now, though.*

The spreadsheet will as usual calculate the change in velocity components from one row to

the ne&t using

v&)tdelta%t* H v&)t* a&)t*delta%t.

To include the drag force in this computation, we might write in 1"


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="15Dg#delta_t D /#"15#$15#delta_t

These formulas are now ready to copy down to fill the table.

xercise! Dou should spend some time playing with this spreadsheetA Bind out how including

air resistance affects the ma&imum range, and the best angle to shoot at for ma&imum range.

4o you thin0 it's greater than or less than 7; degreesK Bind out. !lso, see how the shape of

the path is altered. >ere's an e&ample$

Incidentally, the pre@Galilean medieval theory of projectiles was that they went pretty much

in a straight line until they'd #used up# their initial momentum, whereupon they dropped right

down. It's easy to see this theory loo0s a lot more plausible with high air resistance. )It's also

of course a picture many people still have @@ when roadrunner runs off the edge of a cliff, he

goes in a straight line until he loo0s down, realizes where he is, then dropsA*

4?3!4 T>8 (698!4(>88TA
http://galileo.phys.virginia.edu/classes/581/Projectile5.xlshttp://galileo.phys.virginia.edu/classes/581/Projectile5.xls

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Using Spreadsheets for Projectile Motion