Projectile Motion

The figure is the standard launch diagram for projectile motion. It shows an $x$–$y$ coordinate system with the origin $O$ at the point of projection. From $O$, a curved parabolic path rises to a maximum height and then falls back to the $x$-axis. The initial velocity vector $\mathbf{v}_0$ is drawn as an arrow from $O$ at an angle $\theta_0$ above the positive $x$-axis. Two perpendicular dashed arrows from the tail of $\mathbf{v}_0$ show its rectangular components: a horizontal component $v_0\cos\theta_0$ along the $x$-axis and a vertical component $v_0\sin\theta_0$ along the $y$-axis. A single downward arrow labelled $\mathbf{a} = -g\,\hat{\mathbf{j}}$ is placed near the path, indicating that the only acceleration is constant gravity acting vertically downward.

The physical idea is that the motion separates cleanly into two independent parts. Horizontally, there is no acceleration, so the $x$-component of velocity stays constant at $v_0\cos\theta_0$. Vertically, the acceleration is constant at $-g$, so the $y$-component of velocity changes uniformly from its initial value $v_0\sin\theta_0$ to zero at the peak, then to negative values on the descent. The parabolic shape of the trajectory is the result of combining this uniform horizontal motion with uniformly accelerated vertical motion.

From this figure the textbook develops the core equations of projectile motion. The position at any time $t$ is given by

The velocity components are

The time of flight $T$ (when $y=0$ again) is $T = \frac{2v_0\sin\theta_0}{g}$, the maximum height $H$ (when $v_y=0$) is $H = \frac{v_0^2\sin^2\theta_0}{2g}$, and the horizontal range $R$ is $R = \frac{v_0^2\sin 2\theta_0}{g}$. …

Fig. 3.17 is a single plot of the projectile’s path — the familiar curved arc — drawn on a standard $x$–$y$ coordinate grid. The horizontal axis is labelled $x$ (range) and the vertical axis $y$ (height). The curve itself is a smooth, symmetric parabola that starts at the origin $(0,0)$, rises to a maximum height, then falls back to the $x$-axis at the landing point.

The key visual elements are the velocity vectors drawn at three distinct moments: at launch, at the highest point, and just before landing. At the launch point, the initial velocity $\mathbf{v}_0$ is shown as an arrow making an angle $\theta_0$ with the horizontal. This arrow is resolved into two perpendicular components: a horizontal component $v_{0x}\hat{\mathbf{i}}$ (pointing right along the $x$-axis) and a vertical component $v_{0y}\hat{\mathbf{j}}$ (pointing upward). The figure makes clear that the horizontal component stays constant throughout the flight — the arrow for $v_{0x}$ is drawn with the same length at every stage.

At the apex of the parabola, the vertical velocity arrow disappears: only the horizontal component $v_{0x}\hat{\mathbf{i}}$ remains, because $v_y = 0$ at the highest point. On the descent, the vertical component reappears but now points downward, labelled $-v_{0y}\hat{\mathbf{j}}$. Just before landing, the velocity vector is again at an angle $-\theta_0$ below the horizontal — the same magnitude as the launch velocity but with the vertical component reversed.

The physical idea the figure teaches is that projectile motion is the superposition of two independent motions: uniform motion along $x$ (no acceleration) and uniformly accelerated motion along $y$ (acceleration $= -g$). The parabolic shape emerges because $y$ is a quadratic function of $x$.

The textbook develops the following central formulas from this figure. The initial velocity components are:

At any time $t$, the position coordinates are:

Eliminating $t$ gives the equation of the trajectory:

which is of the form $y = ax - bx^2$ — a parabola. Here $g$ is the acceleration due to gravity ($9.8\ \text{m/s}^2$), $v_0$ is the initial speed, and $\theta_0$ is the launch angle measured from the horizontal.