시간 of Flight 공식: 포물선 운동 (T, Range & 키)

Find time of flight with T = 2v sinθ/g, then range and max height. 풀이 예제s plus a free projectile motion calculator for level-ground launches.

What 포물선 운동 Asks

A projectile is launched with 속력 v at angle θ to the horizontal and then moves under gravity alone. Intro physics ignores air 저항 and usually lands at the same height it started. Horizontal 속도 stays constant. Vertical 속도 behaves like one-dimensional free fall.

If you already have v and θ and only need 확인ed T, R, and H, open the free projectile motion calculator first, then return here for time-of-flight steps and traps.

시간 of Flight 공식

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.

  • 시간 to peak: t_up = vy / g = v sinθ / g.
  • Full flight (same height): T = 2 t_up = 2 v sinθ / g.
  • At the peak, vertical 속도 is zero; horizontal 속도 is still vx.

    • If the landing is higher or lower than the launch (a cliff or raised platform), do not use this T. 풀기 Δy = vy t − ½ g t² for t, then x = vx t.

    풀이 방법 for 시간 in 포물선 운동

  • 변환 속력 to m/s and keep θ in degrees for the calculator (or use radian mode consistently by hand).
  • Find vy = v sinθ.
  • If launch height = landing height, T = 2 vy / g.
  • Peak time is T/2; that is when height is maximum.

    • The vertical part is constant-가속도 kinematics. Refresh SUVAT choice in how to choose the right kinematics equation.

    Range and Maximum 키

  • Range: R = v² sin(2θ) / g = vx T.
  • Maximum height: H = v² sin²θ / (2g) = vy² / (2g).

    • 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.

    단위

  • 속력 v → m/s (convert km/h first: divide by 3.6).
  • 각도 θ → degrees in the calculator; use radians only if your device is in radian mode.
  • g → m/s² (9.8 or 9.81).
  • T → s. R and H → m.

    • 풀이 예제 (시간 of Flight First)

    예제 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.

    예제 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 속력 at 60° keeps R but raises T and H.

    예제 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 공식s Fail

  • Landing on a cliff or a raised platform (Δy ≠ 0).
  • Air 저항, spin, or a strong wind.
  • A moving launch platform unless you include that extra 속도.

    • 흔한 실수

  • Using sinθ in the range 공식 instead of sin(2θ).
  • Forgetting to convert km/h to m/s.
  • Mixing degree and radian mode.
  • Treating range as the distance along the curve instead of the horizontal landing distance.
  • Using T = 2v sinθ/g when the ball lands at a different height.

    • 확인 It on the 계산기

    Projectile motion calculator.

    운동학 calculator.

    속도 calculator.

    학습 time of flight first, then 확인 R and H for every level-ground launch.

    문서

    • 탄성 위치 에너지 공식 (U = ½kx²): 계산기 & 예제
    • 계산 방법 전기 전력 (P = VI, I²R, V²/R)
    • 계산 방법 파동 속도 (v = fλ): 주파수 × 파장
    • 계산 방법 무게 (W = mg): 질량 vs 힘
    • 계산 방법 밀도 (ρ = m/V): 질량 ÷ Volume
    • 계산 방법 가속도 (a = (v − u)/t)

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