One speed, two directions
A launcher is 1.25 m above level ground. Fired horizontally, a ball lands 5.0 m from the point directly below the launcher. Ignore air resistance and use g = 10 m/s^2. The same launcher speed is then aimed vertically upward. How high above the launcher does the ball rise?
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5 m
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Time → speed → vertical rise
Resolve the horizontal flight in time, recover the common launch speed, then use vertical kinematics.
- From 1.25 = (1/2)(10)t^2, obtain t = 0.5 s.
- From R = ut, obtain u = 5.0/0.5 = 10 m/s.
- At the top of the vertical launch, 0 = u^2 - 2gH, so H = 5.0 m.
Trajectory / direct elimination
Use the Cartesian trajectory relation to eliminate flight time; its strongest variant eliminates launch speed as well and links the givens directly to the requested rise.
- For the horizontal launch, h = gR^2/(2u^2).
- For the vertical launch, H = u^2/(2g).
- Eliminating u and g gives hH = R^2/4, hence H = R^2/(4h) = 5.0 m.
Recover speed, then use energy
Recover the common launch speed, then replace the vertical kinematic step with conservation of mechanical energy.
- The horizontal fall gives t = 0.5 s and u = 10 m/s.
- For the vertical launch, (1/2)mu^2 = mgH.
- Cancel m: H = u^2/(2g) = 5.0 m.
Problem renderings
English
A launcher is 1.25 m above level ground. Fired horizontally, a ball lands 5.0 m from the point directly below the launcher. Ignore air resistance and use g = 10 m/s^2. The same launcher speed is then aimed vertically upward. How high above the launcher does the ball rise?
Português (Brasil)
Um lançador está 1,25 m acima de um terreno plano. Quando uma bola é lançada horizontalmente, ela atinge o solo a 5,0 m do ponto diretamente abaixo do lançador. Despreze a resistência do ar e use g = 10 m/s². Em seguida, usando a mesma velocidade de lançamento, a bola é lançada verticalmente para cima. Que altura ela atinge acima do lançador?
简体中文
一个发射器位于水平地面上方1.25米处。小球水平发射后，落地点距离发射器正下方的点5.0米。忽略空气阻力，取g = 10 m/s^2。随后保持发射速率不变，将小球竖直向上发射。小球相对发射器能上升多高？
हिन्दी
एक प्रक्षेपक समतल जमीन से 1.25 m ऊपर है। क्षैतिज रूप से छोड़ी गई गेंद प्रक्षेपक के ठीक नीचे वाले बिंदु से 5.0 m दूर जमीन पर गिरती है। वायु प्रतिरोध की उपेक्षा करें और g = 10 m/s^2 लें। फिर उसी प्रारंभिक चाल से गेंद को सीधा ऊपर छोड़ा जाता है। गेंद प्रक्षेपक से कितनी ऊँचाई तक ऊपर उठेगी?