investigate and analyse theoretically and practically the motion of projectiles near Earth's surface including a qualitative description of the effects of air resistance

What this dot point is asking

VCAA wants you to model a projectile (an object moving only under gravity) by splitting its velocity into independent horizontal and vertical components, then applying the equations of motion to each axis. You also need to describe qualitatively how air resistance changes the trajectory.

The answer

A projectile is any object in flight that is subject only to gravity (we ignore air resistance for the calculations). The trick is that horizontal and vertical motions are independent, linked only by the shared time of flight.

Resolving the initial velocity

If a projectile is launched with speed $v_0$ at angle $\theta$ above the horizontal:

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Horizontal motion

No horizontal force acts (air resistance ignored), so horizontal velocity is constant.

Vertical motion

The only acceleration is gravity, $a_y = -g = -9.8$ m/s squared (taking up as positive). Use SUVAT:

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Key features of the trajectory

The path is a parabola. At maximum height $v_y = 0$, so $h_{\max} = \frac{v_{0y}^2}{2g}$.

For a projectile launched from and landing at the same height, the time of flight is $t = \frac{2 v_{0y}}{g}$ and the range is:

Range is maximised at $\theta = 45°$ on level ground, and complementary launch angles (for example $30°$ and $60°$) give the same range.

Air resistance (qualitative)

Air resistance (drag) acts opposite to the velocity vector at every instant and increases with speed (roughly $F_{drag} \propto v^2$). Its effects on a projectile:

• Range falls. Horizontal velocity decreases throughout the flight rather than staying constant.
• Maximum height falls. Drag removes kinetic energy on the way up.
• Trajectory loses symmetry. The descent is steeper than the ascent because the ball has less horizontal velocity by the time it falls.
• Optimum launch angle is below 45 degrees for maximum range with drag (because the ball benefits from spending less time in the air).

VCAA expects qualitative description only. No numerical drag questions.

Worked example with numbers

A ball is kicked from ground level at $v_0 = 20$ m/s at $\theta = 40°$ above horizontal. Find the maximum height, time of flight, and range.

Resolve: $v_{0x} = 20 \cos 40° = 15.32$ m/s; $v_{0y} = 20 \sin 40° = 12.86$ m/s.

Maximum height: $h_{\max} = \frac{v_{0y}^2}{2g} = \frac{12.86^2}{19.6} = 8.44$ m.

Time of flight: $t = \frac{2 v_{0y}}{g} = \frac{2 \times 12.86}{9.8} = 2.62$ s.

Range: $R = v_{0x} t = 15.32 \times 2.62 = 40.2$ m.

Try it: Projectile motion calculator - enter launch speed, angle and height and get the range, max height and trajectory.

Common traps

Mixing horizontal and vertical equations. Horizontal velocity is constant (no drag). Vertical velocity changes by $9.8$ m/s each second. Two separate columns of working.

Forgetting the sign of $g$. Pick a positive direction (up or down) and apply it consistently to $v_{0y}$, $a_y$ and $y$.

Using the launch speed instead of a component. $v_0 = 25$ m/s at $40°$ does not mean horizontal velocity is $25$ m/s. Resolve first.

Treating a horizontally thrown object as having $v_{0y} = v_0$. If a stone is thrown horizontally off a cliff, $v_{0y} = 0$.

Treating air resistance as a constant force. Drag depends on speed; expect a qualitative description, not a numerical one.

In one sentence

Projectile motion is solved by splitting the launch velocity into horizontal and vertical components and applying constant-velocity equations horizontally and SUVAT vertically with $g = 9.8$ m/s squared, linked by the shared time of flight, with air resistance qualitatively reducing both range and maximum height.