DP Mathematics: Applications and Interpretation Questionbank

Write the differential equation as a system of coupled first order differential equations.

When $t=0$, $x=\dot{x}=0$

Use Euler’s method with a step length of $0.1$ to find an estimate for the value of the displacement and velocity of the particle when $t=1$.

Use the substitution $y=\frac{dx}{dt}$ to write the differential equation as a system of coupled, first order differential equations.

Use Euler’s method with a step length of $0.1$ to find the displacement of the particle when $t=1$.

Find the long-term velocity of the particle.

Use your answers to parts (d), (e) and (f) to comment on the accuracy of the Euler approximation to this model.

By substituting $v=\frac{\text{d}x}{\text{d}t}$ into the equation, find an expression for the velocity of the particle at time $t$. Give your answer in the form $v=f(t)$.

By repeated application of Euler’s method, find an approximation for the terminal velocity, to five significant figures.

Write down the differential equation as a system of first order differential equations.

Use Euler’s method, with a step length of 0.2, to find the displacement and velocity of the object when $t=0.6$.

Use the differential equation to find the terminal velocity for the object.

From your solution to part (a), or otherwise, find the terminal velocity of the object predicted by this model.