# MA 227 Standard S04

Kinematics and Ideal Projectile Motion

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)