a plane travelling horizontally at 150 m/s drops a package from an altitude of 1.3 km. how far horizontally did the package travel?
A plane traveling horizontally at 150 m/s drops a package from 1.3 km altitude. The package travels horizontally about 1,620 meters before hitting the ground.
This classic projectile motion problem separates horizontal and vertical components, as there's no horizontal acceleration but gravity acts downward.
Key Assumptions
- Acceleration due to gravity: g=9.8 m/s2g=9.8,\mathrm{m/s^2}g=9.8m/s2.
- No air resistance.
- Package inherits the plane's horizontal velocity (150 m/s) upon release; initial vertical velocity is 0 m/s.
- Altitude: 1,300 m (converted from 1.3 km).
Step 1: Time of Fall
Vertical motion is free fall:
h=12gt2h=\frac{1}{2}gt^2h=21gt2
1300=12(9.8)t21300=\frac{1}{2}(9.8)t^21300=21(9.8)t2
t2=26009.8≈265.306t^2=\frac{2600}{9.8}\approx 265.306t2=9.82600≈265.306
t≈16.29 st\approx 16.29,\mathrm{s}t≈16.29s.
Step 2: Horizontal Distance
Horizontal motion is constant velocity:
dx=vxt=150×16.29≈2443.5 md_x=v_xt=150\times 16.29\approx
2443.5,\mathrm{m}dx=vxt=150×16.29≈2443.5m No, wait—recalculate precisely:
t=2×13009.8=265.3061=16.29 st=\sqrt{\frac{2\times
1300}{9.8}}=\sqrt{265.3061}=16.29,\mathrm{s}t=9.82×1300=265.3061=16.29s
dx=150×16.29=2,443.5 md_x=150\times
16.29=2,443.5,\mathrm{m}dx=150×16.29=2,443.5m? Error in initial summary.
Corrected precise calc:
t=2hg=26009.8=265.306122≈16.289
st=\sqrt{\frac{2h}{g}}=\sqrt{\frac{2600}{9.8}}=\sqrt{265.306122}\approx
16.289,\mathrm{s}t=g2h=9.82600=265.306122≈16.289s
150×16.289=2,443.35 m150\times
16.289=2,443.35,\mathrm{m}150×16.289=2,443.35m. But similar problems use g=10
for round numbers; with g=9.8 it's ~2.44 km.
Standard forum consensus rounds g=9.8, distance ~2,440 m.
Quick Comparison Table
Parameter| Value| Equation Used 1
---|---|---
Time to ground| 16.3 s| t=2h/gt=\sqrt{2h/g}t=2h/g
Horizontal distance| 2,440 m| d=vtd=vtd=vt
Vertical velocity at impact| 160 m/s| vy=gtv_y=gtvy=gt
Real-World Context
In reality, air resistance reduces both time and distance slightly, but this ideal model is standard for physics problems. Pilots in WWII supply drops aimed ~2-3 km ahead for similar speeds/heights.
TL;DR: ~2,440 meters horizontally.
Information gathered from public forums or data available on the internet and portrayed here.