高度-时间
读数
发生了什么
水平运动和竖直运动互不影响。重力只把竖直速度往下拉，所以轨迹是一条抛物线。打开空气阻力后，与速度方向相反的阻力会缩短射程，让轨迹变得不对称。最近三条轨迹会保留在屏幕上，方便你直接对比不同设置。
- Horizontal positionx = v cos(θ) t
- Vertical positiony = v sin(θ) t − ½gt²
考试大纲
- A-Level Physics and A-Level Maths mechanics AQA 3.4.1.5, OCR, Edexcelprojectiles
- GCSE Physicsindependence of horizontal and vertical motion
挑战
先预测：速度固定时，哪个角度的射程最远？然后打开空气阻力：最佳角度还是 45°，还是会变小？发射看看。
没有空气阻力时，45° 的射程最远；打开空气阻力后，飞行时间越长损失越大，所以最佳角度会低于 45°。“最远射程”那条线会显示你这个球的最佳角度。
系统地学
GCSE Combined and Separate Science 中讲这个主题的课：先看分步讲解，再做考试风格的题目，每次数字都不同。 课程为英文。
常见问题
Why is 45° the best angle?
On level ground with no air resistance the range is v² sin(2θ)/g, which is largest when 2θ = 90°. Launching from a height, or turning drag on, moves the best angle below 45°.
How do you work out the range of a projectile?
Split the launch velocity into v cos θ across and v sin θ up. The time of flight on level ground is 2v sin θ/g, and the range is the horizontal speed times that time.
What does air resistance change?
Drag opposes the velocity, so the ball slows down sideways and the descent is steeper than the climb. Range and peak height both drop, and the path is no longer a parabola.
What angle do I launch at to hit something a set distance away?
On level ground with no air, sin 2θ = gR/v², which has two answers. At 20 m/s, to land 30 m away: sin 2θ = 9.81 × 30 / 400 = 0.736, so θ = 23.7° or 66.3°, a flat shot or a lob. If gR/v² is above 1 it is out of reach. Pick Angles for a range under Solve for to see each step, from a height too.