At the end of the course, each student should be able to…
- S04: Kinematics. Compute and apply position, velocity, and acceleration vector functions.
S04: Kinematics and Ideal Projectile Motion
- If \(\vect r(t)\) represents the position of a particle at time \(t\), then \(\vect v(t)=\vect r’(t)\) is its velocity and \(\vect a(t)=\vect r^{\prime\prime}(t)\) is its acceleration.
- Assuming ideal projectile motion (the only force on a particle is gravity)
in the plane,
the position of a projectile fired with initial position \(\vect {r_0}\)
and initial velocity \(\vect{v_0}\)
after \(t\) units of time have elapsed is
\(\vect r(t)=\vect{r_0}+ \vect{v_0}t-\frac{1}{2}g\vecj t^2\).
- Expanding \(\vect{r_0}=\<x_0,y_0\>\) and \(\vect{v_0}=\<v_0\cos\alpha,v_0\sin\alpha\>\) where \(\alpha\) is the angle of initial launch and \(v_0=\|\vect{v_0}\|\), this may be written as \(\vect r(t)= \<x_0+(v_0\cos\alpha)t,y_0+(v_0\sin\alpha)t-\frac{1}{2}gt^2\>\).
- Formulas such as \(y_{max}=\frac{(v_0\sin\alpha)^2}{2g}\) for the total flight time of a projectile fired from the ground may be derived from the ideal projectile motion position function.
Textbook References
- University Calculus: Early Transcendentals (3rd Ed)
- 12.1 (exercises 1-18)
- 12.2 (exercises 17-23,27,29)