Constant & Variable Velocity (DP IB Applications & Interpretation (AI): HL): Revision Note

Vectors & constant velocity

How can I model motion with constant velocity using vectors?

• The formula for the position vector of an object is $\mathit{r}={\mathit{r}}_0+\mathit{v}t$

  • ${\mathit{r}}_0$ is the position vector of the starting point

  • $\mathit{v}$ is the velocity for the constant velocity

  • $t$ is the time since the object first left the starting point

• An object moves in a straight line if the velocity is constant

Vectors & variable velocity

How can I model motion with variable velocity using vectors?

• Variable velocity is represented by a vector whose components are functions of time

  • $\mathit{v}=\left(\genfrac{}{}{0pt}{}{v_1\left(t\right)}{v_2\left(t\right)}\right)$

• This means the position vector and acceleration vector are also functions of time

  • $\mathit{r}=\left(\genfrac{}{}{0pt}{}{r_1\left(t\right)}{r_2\left(t\right)}\right)$

  • $\mathit{a}=\left(\genfrac{}{}{0pt}{}{a_1\left(t\right)}{a_2\left(t\right)}\right)$

What is the connection between displacement and velocity vectors?

• Velocity is the rate of change of the position

  • $\mathit{v}=\frac{d\mathit{r}}{dt}$

• Differentiate the position vector to find an expression for the velocity vector

  • $\left(\genfrac{}{}{0pt}{}{v_1\left(t\right)}{v_2\left(t\right)}\right)=\left(\begin{array}{c}{r}_1\text{'}\left(t\right)\\ r_2\text{'}\left(t\right)\end{array}\right)$

• Integrate the velocity vector to find an expression for the position vector

  • $\left(\genfrac{}{}{0pt}{}{r_1\left(t\right)}{r_2\left(t\right)}\right)=\left(\begin{array}{c}\int v_{1}dt\\ \int v_{2}dt\end{array}\right)$

What is the connection between acceleration and velocity vectors?

• Acceleration is the rate of change of the velocity

  • $\mathit{a}=\frac{d\mathit{v}}{dt}$

• This means it is the second derivative of the position

  • $\mathit{a}=\frac{d^2\mathit{r}}{d{t}^2}$

• Differentiate the velocity vector to find an expression for the acceleration vector

  • $\left(\genfrac{}{}{0pt}{}{a_1\left(t\right)}{a_2\left(t\right)}\right)=\left(\begin{array}{c}{v}_1\text{'}\left(t\right)\\ v_2\text{'}\left(t\right)\end{array}\right)$

• Differentiate the position vector twice to find an expression for the acceleration vector

  • $\left(\genfrac{}{}{0pt}{}{a_1\left(t\right)}{a_2\left(t\right)}\right)=\left(\begin{array}{c}{r}_1\text{'}\left(t\right)\\ r_2\text{'}\left(t\right)\end{array}\right)$

• Integrate the acceleration vector to find an expression for the velocity vector

  • $\left(\genfrac{}{}{0pt}{}{v_1\left(t\right)}{v_2\left(t\right)}\right)=\left(\begin{array}{c}\int a_{1}dt\\ \int a_{2}dt\end{array}\right)$