26.3 Content
26.3.1 Model and Assumptions
A projectile is any object moving through the air under the influence of gravity alone.
- Constant vertical acceleration: \(g = 9.8\ \text{m/s}^2\) downward
- Zero air resistance: no drag forces
- Independence of motion: horizontal and vertical motions do not affect each other
26.3.2 Interactive: Projectile Trajectory
Explore how launch angle and speed affect projectile motion:
26.3.3 Component Analysis
Split the initial velocity into horizontal and vertical components:
\[v_x = v_0 \cos(\theta) \quad \text{(constant)}\] \[v_{y0} = v_0 \sin(\theta) \quad \text{(changes with time)}\]
Horizontal motion (constant velocity): \[x = v_x t\]
Vertical motion (constant acceleration): \[v_y = v_{y0} - gt\] \[y = v_{y0}t - \frac{1}{2}gt^2\]
26.3.4 Time of Flight (Level Launch)
For a projectile launched and landing at the same height:
\[T = \frac{2v_0 \sin(\theta)}{g}\]
Time of flight depends only on the vertical component of velocity. The horizontal velocity has no effect on how long the projectile stays in the air.
26.3.5 Maximum Height
The projectile reaches maximum height when \(v_y = 0\):
\[H = \frac{v_0^2 \sin^2(\theta)}{2g}\]
26.3.6 Interactive: Maximum Height Analysis
Key observation: At maximum height, the vertical velocity is zero but horizontal velocity remains constant.
26.3.7 Horizontal Range (Level Launch)
For level launch and landing:
\[R = \frac{v_0^2 \sin(2\theta)}{g}\]
Maximum range occurs at \(\theta = 45°\). Complementary angles (e.g., 30° and 60°) give the same range.
26.3.8 Interactive: Range Comparison
Compare trajectories at different launch angles:
26.3.9 Non-Level Launch
When launch and landing heights differ, use the general vertical equation:
\[y = v_{y0}t - \frac{1}{2}gt^2\]
Set \(y\) to the final height (relative to launch) and solve for time.