Please show steps. 13. (1 point) The velocity of a body is given as v(t) 5e st 3

Question

Please show steps.

13. (1 point) The velocity of a body is given as v(t) 5e st 3t, where t is in s (second), v is in m/s. Using Euler's method with a step size of 1 seconds, the distance traveled by the body from t 1s to t 3s is estimated most nearly as Chint 1: velocity is the first derivative of position, hint 2: initial condition here does not correspond to t 0s) a 3.752 m b) 6.261 m c) 8.137 m d) 12.08 m 4. (1 point) The velocity of a body is given as v(t) 5e st 3t, where t is in s (second), v is in m/s. Using Runge-Kutta 4th order method with a step size of 1 seconds, the distance traveled by the body from t 1s to t 3s is estimated most nearly as (hint 1: velocity is the first derivative of position; hint 2: initial condition here does not correspond to t 0s)

Explanation / Answer

5e^(-3t) + 3t = velocity = v(t) = dS/dt = f(t,S) (where t = time and S = distance).

Euler's method states that yn+1=yn+hf(tn,yn), where tn+1=tn+h

We have that h=1, t0=0, y0=0, f(t,y)=3t+5e3t

Let S0 be the start location. So, S(0) = 0 + C with t0 = 1 and h = 1(step size)

S1 = S0 + f(S0,t0) * h

S1 = 0 + C + (5e^(-3*1) + 3*1) * 1

S1 = 3.24893534184 + C

t1 = t0 + h

t1 = 1 + 1 = 2

S2 = S1 + f(S1,t1) * h ; where t1=2 ; S1 = 3.24893534184

S2 = 3.24893534184 + C + (5e^(-3*2) + 3*2) * 1

S2 = 3.24893534184 + C + 6.01239376088

S2 = 9.26132910272 + C

S2 - S0 = S3 - S1 = 9.26132910272 + C - 0 - C = 9.26132910272

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