Consider the motion of a particle of mass m falling vertically under the earth’s gravitational field, and suppose the
downward motion is opposed by a frictional force p(v) dependent on the velocity v(t) of the particle. Then the
velocity satisfies the equation
mv0(t) = mg+ p(v); t � 0
Let m = 1kg, g = 9:8m=sec2, and v(0) = 0. Using Euler’s, Heuns’, and Classical Runge-Kutta methods, solve the
differential equation for 0 � t � 20 and for the following choices of p(v):
1. p(v) = 0:1v
2. p(v) = 0:1v2
Find the answers to at least four digits accuracy. Graph the function v(t). Compare the solutions.
----Answer is not available----
To solve the given differential equation, we need to use Euler's method, Heun's method, and classical Runge-Kutta method. These numerical methods approximate the solution to a differential equation by incrementally calculating the values of the unknown function at different points in time.
Let's start by analyzing the given equation. We have:
m * v'(t) = -m * g + p(v)
where m = 1 kg and g = 9.8 m/sec^2.
Now, let's consider the two different cases for p(v) and solve the differential equation using the numerical methods.
Case 1: p(v) = -0.1 * v
Substituting this into the equation gives us:
v'(t) = -g - 0.1 * v
Case 2: p(v) = 0.1 * v^2
Substituting this into the equation gives us:
v'(t) = -g + 0.1 * v^2
Now, we can proceed with solving the differential equation using Euler's method, Heun's method, and classical Runge-Kutta method. These methods involve using iterative calculations to approximate the solution at different time steps.
1. Euler's Method:
Euler's method approximates the derivative as a difference quotient and uses this to update the function value at each time step.
For both cases, the update rule is given as:
v(t + ∆t) = v(t) + ∆t * v'(t)
where ∆t is the time step size.
2. Heun's Method:
Heun's method predicts the next value of the unknown function using Euler's method, then calculates an average of the predicted and current values to get a better approximation.
For both cases:
v(t + ∆t) = v(t) + ∆t * [v'(t) + v'(t + ∆t)] / 2
3. Classical Runge-Kutta Method:
The classical Runge-Kutta method is a higher-order method that calculates the weighted average of several estimates to improve accuracy.
For both cases:
k1 = ∆t * v'(t)
k2 = ∆t * v'(t + ∆t/2)
k3 = ∆t * v'(t + ∆t/2)
k4 = ∆t * v'(t + ∆t)
v(t + ∆t) = v(t) + (k1 + 2k2 + 2k3 + k4) / 6
By applying these update rules iteratively, we can approximate the solution v(t) for 0 ≤ t ≤ 20. After obtaining the numerical solutions, we can plot the function v(t) to compare the results from the different methods.
Note: To ensure four-digit accuracy, it may be necessary to use a smaller time step (∆t) and/or iterate over a larger number of time steps.