What it shows
For a projectile launched at the same speed from ground level, the range is proportional to sin 2 theta, which is greatest at 45 degrees. Equal ranges occur for angles that add up to 90 degrees.
The science
With launch speed v at angle theta the horizontal speed is v cos theta and the time in flight is 2 v sin theta / g. So the range is R = v squared sin 2 theta / g. Since sin 2 theta is greatest when 2 theta = 90 degrees, the maximum range is at 45 degrees. For two angles theta and 90 minus theta, sin 2 theta is the same, so the ranges are equal. In a real launch air drag and the launch height above ground move the best angle a little below 45 degrees.
Materials
Cost: under Rs 50. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A ruler catapult: a ruler pivoted on a pencil with a rubber band, or a simple spring launcher
- A soft foam or paper ball
- A protractor and a cardboard ramp to set angles
- A tape measure on the floor
- Chalk
- A table of angle and range
- Graph paper and a calculator
Safety
For this project: Use only soft balls and launch in an open area with no one in front. Wear eye protection and never aim at anyone.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Make the launcher so that the same stretch of the rubber band is used each time. Mark the pull-back position with tape.
Fix the launcher on a base marked with angles of 15, 25, 35, 45, 55, 65 and 75 degrees using the protractor.
Mark a clear landing area with a tape measure along the floor and chalk the starting line under the launch point.
Fire the ball three times at each angle and mark where it first touches the floor. Take the average range for each angle.
Write the angles and mean ranges in a table and add a third column for sin 2 theta calculated with a calculator.
Plot range against angle and draw a smooth curve through the points. Mark the highest point.
Plot range against sin 2 theta, which should give a straight line through the origin, and find its slope.
Check the pairs of complementary angles, 25 and 65, and 15 and 75, and compare their ranges.
What you should see
The range rises to a peak at about 40 to 45 degrees and falls again, complementary angles give nearly equal ranges, and range against sin 2 theta is close to a straight line.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1What is the formula for the range of a projectile on level ground?
R = v squared sin 2 theta / g.
2Why do 30 and 60 degrees give the same range?
sin 60 and sin 120 are equal, so sin 2 theta is the same.
3Why is the best angle a little under 45 degrees in your test?
The ball is launched from above ground level and air resistance reduces the range.
4What does the slope of the R against sin 2 theta graph give?
The value of v squared divided by g, from which you can find the launch speed.
Take it further
Use the slope to calculate the launch speed and check it by timing the ball over a short flat distance.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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