Time of Flight الصيغة: حركة المقذوفات (T, Range & الطول)
Find time of flight with T = 2v sinθ/g, then range and max height. مثال محلولs plus a free projectile motion calculator for level-ground launches.
What حركة المقذوفات Asks
A projectile is launched with speed v at angle θ to the horizontal and then moves under gravity alone. Intro physics ignores air resistance and usually lands at the same height it started. Horizontal velocity stays constant. Vertical velocity behaves like one-dimensional free fall.
If you already have v and θ and only need checked T, R, and H, open the free projectile motion calculator first, then return here for time-of-flight steps and traps.
Time of Flight الصيغة
On level ground, time of flight (hang time) is T = 2 v sinθ / g. With components vx = v cosθ and vy = v sinθ, that is the same as T = 2 vy / g — twice the time to the peak.
If the landing is higher or lower than the launch (a cliff or raised platform), do not use this T. Solve Δy = vy t − ½ g t² for t, then x = vx t.
كيفية الحل for Time in حركة المقذوفات
The vertical part is constant-acceleration kinematics. Refresh SUVAT choice in how to choose the right kinematics equation.
Range and Maximum الطول
sin(2θ) peaks at 2θ = 90°, so θ = 45° gives maximum range on level ground. Complementary angles such as 30° and 60° share the same range, but not the same hang time or peak height.
Units
الشغلed Examples (Time of Flight First)
Example 1: Find T for 20 m/s at 45°
g = 9.8. vy = 20 sin45° ≈ 14.14 m/s. T = 2(14.14) / 9.8 ≈ 2.886 s. Peak at ≈ 1.443 s. Then R ≈ 40.82 m and H ≈ 10.20 m.
Example 2: Find time for 30 m/s at 30°
vy = 30 × 0.5 = 15 m/s. T = 2(15) / 9.8 ≈ 3.061 s. R ≈ 79.55 m, H ≈ 11.48 m. Same speed at 60° keeps R but raises T and H.
Example 3: 10 m/s straight up
θ = 90°. R = 0, T = 2(10)/9.8 ≈ 2.041 s, H ≈ 5.10 m — same as 1D free-fall up and down.
When These الصيغةs Fail
أخطاء شائعة
Check It on the Calculator
Learn time of flight first, then check R and H for every level-ground launch.