Question

Problem 4.3. The drag (air resistance) experienced by airplanes can be simply modeled using a cubic equation in terms of velocity, D(v) = 0o + ajv + a202 + azva, where v is the velocity of the plane (we will ignore units in this problem and D is the drag. Engineers working in a wind tunnel test a plane at different velocities and record the drag in the following table: v | 150 175 200 250 300 400 D 7424 10758 16706 30991 53915 127707 Calculate the least squares cubic equation D(U) for this data.

Answer 1

Using least square method we obtain the required equation.

D (9) = + 4 9 + 0 0 + 9, 9 , (64, D. = D(95) , 1, 2, - , 6 , Using least square method, we have, 2, 4, 4 6 1 4 4 4 4 Dys, = 45 14 15 16 2 4 4 1 4 0 1 0 2 4 4 14 4 1 4 1 4 ๑HE ME -- เS ( ( H A Nๆ งNE เ |

A H B C D Ε F к v o v^2 v^3 v^4 V^5 V^6 Dv Dv^2 Dv^3 150 7424 22500 3375000506250000 75937500000 11390625000000 1113600 167040000 25056000000 175 10758 30625 5359375 937890625 164130859375 28722900390625 1882650 329463750 57656156250 200 16706 40000 8000000 1600000000 320000000000 64000000000000 3341200 668240000 133648000000 250 30991 62500 15625000 3906250000 976562500000 244140625000000 7747750 1936937500 484234375000 300 53915 90000 27000000 8100000000 2430000000000 729000000000000 16174500 4852350000 1455705000000 400 127707 160000 64000000 25600000000 10240000000000 4096000000000000 51082800 20433120000 8173248000000 1475 247501 405625 123359375 40650390625 14206630859375 5173254150390620 81342500 28387151250 10329547531250 Sum

$\small \\ \text{Using above table we have,} \\\\247501=6a_0+1475a_1+ 405625a_2+123359375a_3 \\\\81342500=1475a_0+ 405625a_1+123359375a_2+40650390625a_3 \\\\28387151250=405625a_0+123359375a_1+ 40650390625a_2+14206630859375a_3 \\\\10329547531250=123359375a_0+40650390625a_1+14206630859375a_2+5173254150390625a_3$

Solving this system, we get,

$\small a_0=2038,~ a_1=-11.875,~ a_2=0.014331,~ a_3=0.0020021$

Therefore required equation is

$\small D(v)=2038-11.875v+0.014331v^2+0.0020021v^3$