Question

Part B - Modelling and Solving First Order ODES Charlie decides to create a theoretical model of his riding velocity to test whether his watch is callibrated properly. To simplify the problem, Charlie decides to test the watch on flat ground. As a further simplification, Charlie decides to start their trial at 10 m/s and then let the bike coast (aka no external force). 1. Given drag force can be modelled with equation Fa = dva, draw a free body diagram of the bike and show that velocity can be modelled with the ODE: du d +-02 = 0. dtm 2. If Charlie linearises the ODE about the initial velocity of 10 m/s, show that the approximate velocity, va, can be modelled with the ODE: dva 2. 200 Va =- 1000 dtm m 3. Solve the linearised ODE for ve given m = 70 kg, d=1. 4. Solve the original (non-linear) ODE for v given m = 70 kg, d=1. 5. Plot the solutions for both the linear and nonlinear ODEs over the interval of one minute. Under what conditions does the linearised ODE accurately model velocity? Hint: You can adjust the y-axis limits of a plot using the ylim function.

Answer 1

Ft = duz - 6666 - Drag force = -ma Lane hewton's by Velocity LLLLLLL du acceleration Riveis Since are for samdy 4 (II) in equilibrium TI - C Point from (2) alt -mdy I du. 2 ldv + d v2=0 I m (2) Since frus is frus= rely - v2fro) continuously differentiable fonction. f(volt flue) (v-voll+ ol 8²) - VO - velocity Valonas given approximated we are dva - mdood - 20d Va m £ d va + 20 I vam + 20 lova - lood dt

- linearized ODF va me 90 kg i dva + 200 va = 1oo d m + - at m 2 + 20 100 X 270 Th I.F = da + 2a = 10 e largest seint Vaelat = to set at to vo exato + va (soetat te) <***] 22+ (4) Non-linear du + du?.0 CHI dt, M . m=90 2+1 + alt fc t- foc .70C