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201:. The effect of air resistance varies enormously depending on the size and geometry of the falling object—for example, the equations are hopelessly wrong for a feather, which has a low mass but offers a large resistance to the air. (In the absence of an atmosphere all objects fall at the same rate, as astronaut
302:
49 m/s (9.8 m/s Ă— 5 s) due to air resistance). Air resistance induces a drag force on any body that falls through any atmosphere other than a perfect vacuum, and this drag force increases with velocity until it equals the gravitational force, leaving the object to fall at a constant
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Based on wind resistance, for example, the terminal velocity of a skydiver in a belly-to-earth (i.e., face down) free-fall position is about 195 km/h (122 mph or 54 m/s). This velocity is the asymptotic limiting value of the acceleration process, because the effective forces on the
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causes the acceleration measured on the rotating surface of the Earth to differ from the acceleration that is measured for a free-falling body: the apparent acceleration in the rotating frame of reference is the total gravity vector minus a small vector toward the north-south axis of the Earth,
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The first equation shows that, after one second, an object will have fallen a distance of 1/2 Ă— 9.8 Ă— 1 = 4.9 m. After two seconds it will have fallen 1/2 Ă— 9.8 Ă— 2 = 19.6 m; and so on. On the other hand, the penultimate equation becomes grossly inaccurate at great distances. If an object fell
228:
An initially stationary object which is allowed to fall freely under gravity falls a distance proportional to the square of the elapsed time. This image, spanning half a second, was captured with a stroboscopic flash at 20 flashes per second. During the first 0.05 s the ball drops one unit of
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In all cases, the body is assumed to start from rest, and air resistance is neglected. Generally, in Earth's atmosphere, all results below will therefore be quite inaccurate after only 5 seconds of fall (at which time an object's velocity will be a little less than the vacuum value of
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negligibly varies with height during the fall (that is, they assume constant acceleration). The last equation is more accurate where significant changes in fractional distance from the centre of the planet during the fall cause significant changes in
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diving down on its prey. The same terminal velocity is reached for a typical .30-06 bullet dropping downwards—when it is returning to earth having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army
Ordnance study.
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is reasonable for objects falling to Earth over the relatively short vertical distances of our everyday experience, but is not valid for greater distances involved in calculating more distant effects, such as spacecraft trajectories.
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Terminal velocity depends on atmospheric drag, the coefficient of drag for the object, the (instantaneous) velocity of the object, and the area presented to the airflow.
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to study rolling balls, the ramp slowing the acceleration enough to measure the time taken for the ball to roll a known distance. He measured elapsed time with a
2089:{\displaystyle t={\frac {{\frac {\pi }{2}}-\arcsin {\Big (}{\sqrt {\frac {x}{r}}}{\Big )}+{\sqrt {{\frac {x}{r}}\ (1-{\frac {x}{r}})}}}{\sqrt {2\mu }}}\,r^{3/2}}
42:
246:, which might be thought of as "metres per second, per second"; or 32.18 ft/s as "feet per second per second") approximately. A coherent set of units for
216:
for example. Nevertheless, they are usually accurate enough for dense and compact objects falling over heights not exceeding the tallest man-made structures.
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of the two bodies. This equation should be used whenever there is a significant difference in the gravitational acceleration during the fall. Note that when
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The equations ignore air resistance, which has a dramatic effect on objects falling an appreciable distance in air, causing them to quickly approach a
2398:(Report) (1999 ed.). U.S. Department of Commerce Technology Administration and National Institute of Standards and Technology. pp. 188–190.
1825:). In this case, the terminal velocity increases to about 320 km/h (200 mph or 90 m/s), which is almost the terminal velocity of the
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Measured fall time of a small steel sphere falling from various heights. The data is in good agreement with the predicted fall time of
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Removing the simplifying assumption of uniform gravitational acceleration provides more accurate results. We find from the
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and equation for universal gravitation (r+d= distance of object above the ground from the center of mass of planet):
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body balance each other more and more closely as the terminal velocity is approached. In this example, a speed of 50
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Newton's
Gravity: An Introductory Guide to the Mechanics of the Universe, Undergraduate Lecture Notes in Physics
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distance (about 12 mm), by 0.10 s it has dropped at total of 4 units, by 0.15 s 9 units, and so on.
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1810:% of terminal velocity is reached after only about 3 seconds, while it takes 8 seconds to reach 90
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1708:{\displaystyle \ v_{i}={\sqrt {2GM{\Big (}{\frac {1}{r}}-{\frac {1}{r+d}}{\Big )}}}\ }
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MacDougal, D.W. (2012). "Chapter 2 - Galileo's Great
Discovery: How Things Fall".
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Higher speeds can be attained if the skydiver pulls in his or her limbs (see also
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at the surface of Earth, but assumes a small distance of fall, so the change in
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is the radius from the falling object to the center of the astronomical body.
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The equations also ignore the rotation of the Earth, failing to describe the
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The following equations start from the general equations of linear motion:
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was the first to demonstrate and then formulate these equations. He used a
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000 m to Earth, then the results of both equations differ by only 0.08
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demonstrated by dropping a hammer and a feather on the surface of the
2311:(the earliest modern investigations of the motion of falling bodies)
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2272:{\displaystyle t={\frac {\pi }{2}}{\sqrt {\frac {r^{3}}{2\mu }}}}
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corresponding to staying stationary in that frame of reference.
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325:. This equation occurs in many applications of basic physics.
18:
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Apart from the last formula, these formulas also assume that
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National
Institute of Standards and Technology Monograph 155
1950:, measured from the centers of the two bodies, is given by:
2389:
From
Sundials to Clocks: Understanding Time and Frequency
1470:{\displaystyle \ v_{i}={\sqrt {\frac {2GMd}{r^{2}}}}\ }
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164 km, then the difference changes to almost 64
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407:{\displaystyle d(t)=d_{0}+v_{0}t+{1 \over 2}at^{2}}
49:. Unsourced material may be challenged and removed.
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1208:{\displaystyle \ v_{a}={\frac {\sqrt {2gd}}{2}}\ }
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142:-bound conditions. Assuming constant acceleration
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1515:of a falling object that has travelled distance
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1133:of a falling object that has travelled distance
903:of a falling object that has travelled distance
1331:, this equation is used for larger radii where
166:by the Earth’s gravitational field of strength
16:Mathematical description of a body in free fall
535:{\displaystyle F=G{{mM} \over {(r+d)^{2}}}=mg}
2418:. New York: Springer Science+Business Media.
2332:, the foundation of the law of falling bodies
754:{\displaystyle \ t=\ {\sqrt {\frac {2d}{g}}}}
280:is measured in metres per second squared, so
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1018:of an object that has been falling for time
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109:Learn how and when to remove this message
1927:formula for radial elliptic trajectories
1088:{\displaystyle \ v_{a}={\frac {1}{2}}gt}
649:{\displaystyle \ d={\frac {1}{2}}gt^{2}}
579:travelled by an object falling for time
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2387:Jespersen, James; Fitz-Randolph, Jane.
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1887:{\displaystyle {\frac {G(M+m)}{r^{2}}}}
973:{\displaystyle \ v_{i}={\sqrt {2gd}}\ }
799:of a falling object after elapsed time
1910:is the mass of the astronomical body,
1916:is the mass of the falling body, and
1593:(used for large fall distances where
687:taken for an object to fall distance
148:Newton's law of universal gravitation
7:
1835:astronomical bodies other than Earth
47:adding citations to reliable sources
233:Near the surface of the Earth, the
2279:, which is the time to collision.
1409:is small and relatively constant:
14:
2441:Falling body equations calculator
2142:standard gravitational parameters
131:of objects subject to a constant
23:
2358:, for a comprehensive study of
1772:is the acceleration of gravity.
162:is the force exerted on a mass
34:needs additional citations for
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1753:{\displaystyle {\sqrt {2h/g}}}
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58:"Equations for a falling body"
1:
459:{\displaystyle v(t)=v_{0}+at}
242: = 9.807 m/s (
1790:%; however, if it fell from
286:must be measured in metres,
2424:10.1007/978-1-4614-5444-1_2
2133:{\displaystyle \mu =G(M+m)}
1619:can change significantly):
235:acceleration due to gravity
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859:{\displaystyle \ v_{i}=gt}
1814:%, 15 seconds to reach 99
1508:{\displaystyle \ v_{i}\ }
1357:is smaller than standard
1246:{\displaystyle \ v_{i}\ }
1126:{\displaystyle \ v_{a}\ }
1011:{\displaystyle \ v_{a}\ }
896:{\displaystyle \ v_{i}\ }
792:{\displaystyle \ v_{i}\ }
244:metres per second squared
2196:, as expected; and when
146:due to Earth’s gravity,
1482:Instantaneous velocity
1220:Instantaneous velocity
870:Instantaneous velocity
766:Instantaneous velocity
270:is essential. Assuming
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1902:gravitational constant
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1541:on a planet with mass
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1279:on a planet with mass
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298:in metres per second.
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2364:Scientific Revolution
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1612:{\displaystyle \ g\ }
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1586:{\displaystyle \ r\ }
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1560:{\displaystyle \ M\ }
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170:. Assuming constant
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43:improve this article
2362:and his times, the
2315:Equations of motion
2215:{\displaystyle x=0}
2189:{\displaystyle t=0}
2163:{\displaystyle x=r}
2330:Mean speed theorem
2302:De motu antiquiora
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1715:
1680:
1631:
1623:
1622:
1595:
1594:
1569:
1568:
1543:
1542:
1517:
1516:
1492:
1484:
1483:
1477:
1451:
1437:
1421:
1413:
1412:
1385:
1384:
1359:
1358:
1333:
1332:
1307:
1306:
1281:
1280:
1255:
1254:
1230:
1222:
1221:
1215:
1171:
1163:
1162:
1135:
1134:
1110:
1102:
1101:
1095:
1056:
1048:
1047:
1020:
1019:
995:
987:
986:
980:
941:
933:
932:
905:
904:
880:
872:
871:
837:
829:
828:
801:
800:
776:
768:
767:
761:
737:
717:
716:
689:
688:
663:
662:
656:
636:
609:
608:
581:
580:
555:
554:
548:
510:
473:
472:
437:
417:
416:
394:
365:
352:
332:
331:
320:
313:
293:
292:in seconds and
287:
281:
275:
265:
259:
253:
247:
237:
222:
214:Coriolis effect
181:
127:describing the
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
2479:
2477:
2469:
2468:
2463:
2458:
2448:
2447:
2444:
2443:
2436:
2435:External links
2433:
2430:
2429:
2401:
2378:
2377:
2375:
2372:
2369:
2368:
2356:Stillman Drake
2346:
2345:
2343:
2340:
2339:
2338:
2333:
2327:
2322:
2317:
2312:
2296:
2293:
2284:
2281:
2264:
2261:
2255:
2251:
2242:
2239:
2234:
2231:
2211:
2208:
2205:
2185:
2182:
2179:
2159:
2156:
2153:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2108:
2097:
2096:
2083:
2079:
2075:
2071:
2063:
2060:
2053:
2048:
2045:
2040:
2037:
2034:
2026:
2023:
2016:
2011:
2003:
2000:
1992:
1987:
1984:
1981:
1976:
1973:
1965:
1962:
1879:
1875:
1870:
1867:
1864:
1861:
1858:
1855:
1778:
1775:
1747:
1743:
1739:
1736:
1717:
1716:
1697:
1689:
1686:
1683:
1679:
1674:
1669:
1666:
1659:
1654:
1651:
1648:
1643:
1638:
1634:
1620:
1605:
1579:
1553:
1527:
1499:
1495:
1479:
1478:
1458:
1454:
1449:
1446:
1443:
1440:
1433:
1428:
1424:
1410:
1395:
1369:
1343:
1317:
1291:
1265:
1237:
1233:
1217:
1216:
1199:
1195:
1192:
1189:
1183:
1178:
1174:
1160:
1145:
1117:
1113:
1097:
1096:
1084:
1081:
1076:
1073:
1068:
1063:
1059:
1045:
1030:
1002:
998:
982:
981:
964:
961:
958:
953:
948:
944:
930:
915:
887:
883:
867:
866:
855:
852:
849:
844:
840:
826:
811:
783:
779:
763:
762:
747:
743:
740:
730:
727:
714:
699:
673:
658:
657:
643:
639:
635:
630:
627:
622:
619:
606:
591:
565:
547:
544:
543:
542:
531:
528:
525:
517:
513:
509:
506:
503:
500:
495:
492:
486:
483:
480:
455:
452:
449:
444:
440:
436:
433:
430:
427:
424:
401:
397:
393:
388:
385:
380:
377:
372:
368:
364:
359:
355:
351:
348:
345:
342:
339:
221:
218:
180:
177:
150:simplifies to
117:
116:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
2478:
2467:
2464:
2462:
2459:
2457:
2454:
2453:
2451:
2442:
2439:
2438:
2434:
2425:
2421:
2414:
2413:
2405:
2402:
2397:
2390:
2383:
2380:
2373:
2365:
2361:
2357:
2351:
2348:
2341:
2337:
2334:
2331:
2328:
2326:
2323:
2321:
2318:
2316:
2313:
2310:
2309:
2304:
2303:
2299:
2298:
2294:
2292:
2289:
2282:
2280:
2262:
2259:
2253:
2249:
2240:
2237:
2232:
2229:
2209:
2206:
2203:
2183:
2180:
2177:
2157:
2154:
2151:
2143:
2124:
2121:
2118:
2112:
2109:
2106:
2081:
2077:
2073:
2069:
2061:
2058:
2046:
2043:
2038:
2035:
2024:
2021:
2014:
2001:
1998:
1985:
1982:
1979:
1974:
1971:
1963:
1960:
1953:
1952:
1951:
1948:
1942:
1936:
1930:
1928:
1923:
1920:
1914:
1908:
1903:
1898:
1877:
1873:
1865:
1862:
1859:
1853:
1841:
1836:
1831:
1828:
1824:
1819:
1818:% and so on.
1803:
1794:, which is 42
1793:
1776:
1770:
1764:
1745:
1741:
1737:
1734:
1723:
1687:
1684:
1681:
1677:
1672:
1667:
1664:
1652:
1649:
1646:
1641:
1636:
1632:
1621:
1603:
1577:
1551:
1525:
1497:
1493:
1481:
1480:
1456:
1452:
1447:
1444:
1441:
1438:
1431:
1426:
1422:
1411:
1393:
1367:
1341:
1315:
1289:
1263:
1235:
1231:
1219:
1218:
1197:
1193:
1190:
1187:
1181:
1176:
1172:
1161:
1143:
1115:
1111:
1099:
1098:
1082:
1079:
1074:
1071:
1066:
1061:
1057:
1046:
1028:
1000:
996:
984:
983:
962:
959:
956:
951:
946:
942:
931:
913:
885:
881:
869:
868:
853:
850:
847:
842:
838:
827:
809:
781:
777:
765:
764:
745:
741:
738:
728:
725:
715:
697:
671:
660:
659:
641:
637:
633:
628:
625:
620:
617:
607:
589:
563:
552:
551:
545:
529:
526:
523:
515:
507:
504:
501:
493:
490:
484:
481:
478:
471:
470:
469:
466:
453:
450:
447:
442:
438:
434:
428:
422:
414:
399:
395:
391:
386:
383:
378:
375:
370:
366:
362:
357:
353:
349:
343:
337:
329:
326:
323:
316:
310:
307:
305:
299:
296:
290:
284:
278:
273:
268:
262:
256:
250:
245:
240:
236:
226:
219:
217:
215:
210:
208:
204:
200:
195:
193:
189:
185:
178:
176:
173:
169:
165:
161:
157:
153:
149:
145:
141:
138:under normal
137:
134:
133:gravitational
130:
126:
125:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
2411:
2404:
2395:
2382:
2350:
2306:
2300:
2286:
2098:
1946:
1944:to a height
1940:
1934:
1931:
1924:
1918:
1912:
1906:
1896:
1839:
1832:
1820:
1804:
1780:
1768:
1762:
467:
415:
330:
327:
321:
314:
311:
308:
300:
294:
288:
282:
276:
266:
260:
254:
248:
238:
232:
211:
196:
182:
171:
167:
163:
159:
155:
151:
143:
129:trajectories
122:
120:
105:
99:October 2017
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
1567:and radius
203:David Scott
192:water clock
2450:Categories
2374:References
1823:freeflying
69:newspapers
2461:Equations
2320:Free fall
2263:μ
2238:π
2222:it gives
2107:μ
2062:μ
2039:−
1986:
1980:−
1972:π
1932:The time
1673:−
553:Distance
546:Equations
124:equations
121:A set of
2295:See also
1760:, where
272:SI units
220:Overview
158:, where
2466:Falling
2456:Gravity
2360:Galileo
2325:Gravity
1900:is the
1816:
1812:
1808:
1800:
1796:
1788:
1784:
1777:Example
184:Galileo
179:History
83:scholar
2099:where
2030:
1983:arcsin
1894:where
1703:
1629:
1607:
1601:
1581:
1575:
1555:
1549:
1529:
1523:
1503:
1490:
1465:
1419:
1397:
1391:
1371:
1365:
1345:
1339:
1319:
1313:
1293:
1287:
1267:
1261:
1241:
1228:
1203:
1169:
1147:
1141:
1121:
1108:
1054:
1032:
1026:
1006:
993:
968:
939:
917:
911:
891:
878:
835:
813:
807:
787:
774:
732:
723:
701:
695:
675:
669:
615:
593:
587:
567:
561:
85:
78:
71:
64:
56:
2416:(PDF)
2392:(PDF)
2342:Notes
661:Time
140:Earth
136:force
90:JSTOR
76:books
2305:and
1833:For
264:and
207:Moon
188:ramp
62:news
2420:doi
1802:%.
209:.)
45:by
2452::
2394:.
1929::
1904:,
1782:10
929::
825::
713::
605::
306:.
274:,
258:,
252:,
156:mg
154:=
2426:.
2422::
2366:.
2260:2
2254:3
2250:r
2241:2
2233:=
2230:t
2210:0
2207:=
2204:x
2184:0
2181:=
2178:t
2158:r
2155:=
2152:x
2128:)
2125:m
2122:+
2119:M
2116:(
2113:G
2110:=
2082:2
2078:/
2074:3
2070:r
2059:2
2052:)
2047:r
2044:x
2036:1
2033:(
2025:r
2022:x
2015:+
2010:)
2002:r
1999:x
1991:(
1975:2
1964:=
1961:t
1947:x
1941:r
1935:t
1919:r
1913:m
1907:M
1897:G
1878:2
1874:r
1869:)
1866:m
1863:+
1860:M
1857:(
1854:G
1840:g
1769:g
1763:h
1746:g
1742:/
1738:h
1735:2
1696:)
1688:d
1685:+
1682:r
1678:1
1668:r
1665:1
1658:(
1653:M
1650:G
1647:2
1642:=
1637:i
1633:v
1604:g
1578:r
1552:M
1526:d
1498:i
1494:v
1457:2
1453:r
1448:d
1445:M
1442:G
1439:2
1432:=
1427:i
1423:v
1394:g
1368:g
1342:g
1316:r
1290:M
1264:d
1236:i
1232:v
1198:2
1194:d
1191:g
1188:2
1182:=
1177:a
1173:v
1144:d
1116:a
1112:v
1083:t
1080:g
1075:2
1072:1
1067:=
1062:a
1058:v
1029:t
1001:a
997:v
963:d
960:g
957:2
952:=
947:i
943:v
914:d
886:i
882:v
854:t
851:g
848:=
843:i
839:v
810:t
782:i
778:v
746:g
742:d
739:2
729:=
726:t
698:d
672:t
642:2
638:t
634:g
629:2
626:1
621:=
618:d
590:t
564:d
530:g
527:m
524:=
516:2
512:)
508:d
505:+
502:r
499:(
494:M
491:m
485:G
482:=
479:F
454:t
451:a
448:+
443:0
439:v
435:=
432:)
429:t
426:(
423:v
400:2
396:t
392:a
387:2
384:1
379:+
376:t
371:0
367:v
363:+
358:0
354:d
350:=
347:)
344:t
341:(
338:d
322:g
315:g
295:v
289:t
283:d
277:g
267:v
261:t
255:d
249:g
239:g
172:g
168:g
164:m
160:F
152:F
144:g
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
