Combining Two Perpendicular Motions
A pen is run straight across a sheet of paper along a ruler while the sheet itself is pulled away at right angles to it. Neither hand moves in a slanted direction, and yet the line left on the paper is slanted. The simulation draws the line as it happens, marks it at equal intervals of time, and lays the two motions out as the sides of a right-angled triangle whose hypotenuse is the line itself.
इस सिमुलेशन का उपयोग कैसे करें
- Set the speed of the pen across the sheet and the speed the sheet is pulled away — both stay live during a run
- Start, pause and reset the run, or play it at half speed, normal speed or double speed
- Turn the velocity vectors, the displacement triangle, the timing marks and the graph ruling on or off
- Watch from the desk, from a low angle, from straight down, or orbit the scene freely
- Record a finished run into the table, then change the speeds and record another
क्या देखें
- The angle measured off the drawn ink and the angle predicted by arctan(v_paper / v_pen) agree to the last decimal place shown
- The marks along the line are equally spaced, and so are the marks on both legs of the triangle — every part of the motion is uniform
- The blue leg is the path the pen really took through the room: dead straight along the ruler, with no slant in it at all
- The length of the line is always √(x² + y²), which is the resultant speed multiplied by the elapsed time
- Hold one speed at zero and the line runs straight across or straight along; the slant appears only when both motions run together
- Move a slider half way through a run and the line kinks at that exact point, then continues straight at the new angle
इसके पीछे की भौतिकी
The two motions are independent: the pen covers the same distance across the sheet whether the sheet is racing away or standing still, and the sheet does the same in return. After a time t the pen has moved x = v_pen·t across the paper and the paper has moved y = v_paper·t along it, so the point of contact has traced, in the paper’s own frame, a straight line of length √(x² + y²) at an angle θ to the pen’s direction given by tan θ = y/x = v_paper / v_pen. The speed of the pen relative to the paper is therefore √(v_pen² + v_paper²) — larger than either motion alone — and because the ratio of the speeds is constant while they are, the angle is constant and the mark is straight. Change one speed part way through and the line bends at that instant, because the slant only ever reports the ratio of the two speeds at the moment the ink was laid down. This is vector addition of velocities in its plainest form, and the same construction that gives a boat crossing a river its drift, or an aircraft its ground track in a crosswind.