PROJECTILE MOTION. At any given point in the motion, the velocity vector is always a tangent to the path.

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Berenice Hawkins
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1 PROJECTILE MOTION A pojectile is any object that has been thown though the ai. A foce must necessaily set the object in motion initially but, while it is moing though the ai, no foce othe than gaity acts on it (we shall ignoe ai esistance fo now). The path, o tajectoy, of the pojectile is paabolic. tajectoy pojectile mg At any gien point in the motion, the elocity ecto is always a tangent to the path. Note also that the ecto mg = weight foce = the only foce acting on the object = net foce Now conside the following diagam: V m H mg = F net The School Fo Excellence 016 The Essentials Physics Book 1 Page 15

2 Notice that the elocity ecto,, has two components: A hoizontal component, H. A etical component V. The net foce, mg, has: An effect on V. No effect on H. Thus, thee is no acceleation hoizontally (which is to say that H emains constant thoughout the motion) but thee is indeed a etical acceleation. The next diagam shows the path taken by a pojectile that has been thown hoizontally. The position of the pojectile is shown at equal time inteals. Notice that it taels at constant elocity hoizontally (fo it coes equal distances in equal time inteals) but it is acceleating etically (it coes geate distances in successie equal time inteals). As one would expect, it is moing at constant speed hoizontally, but it is speeding up etically. The School Fo Excellence 016 The Essentials Physics Book 1 Page 16

3 The diagam that follows shows how the elocity ecto changes as the pojectile moes along its tajectoy. Also shown ae the hoizontal and etical components of the elocity ecto. Of couse, the hoizontal component stays constant but the etical component changes. The School Fo Excellence 016 The Essentials Physics Book 1 Page 17

4 We usually handle pojectile motion poblems by beaking up the motion into hoizontal and etical components. Fo the hoizontal component, we use: d = t Fo the etical component, we use the constant acceleation fomulae: = u + d = ut + at 1 at d = t 1 at u + d = t = u + ad EXAMPLE 8 A tennis ball is pojected at an initial elocity of 100 ms -1 at an angle of eleation of 30 fom the top of a 10 m high towe. Calculate: (a) (b) (c) (d) (e) The time taken to each maximum height. The maximum height aboe gound eached. The time of flight. The hoizontal distance fom the base of the towe to the point whee the featheed ceatue hits the gound. The elocity with which the citte hits the gound. Solution It would be athe nice to stat with a diagam: The School Fo Excellence 016 The Essentials Physics Book 1 Page 18

5 We will need to know the hoizontal and etical components of the initial elocity. Some simple tigonomety takes cae of that: 100sin30 = 50 m/s 100 m/s (a) u = +50 m / s a = 10 m / s cos30 = 86.6 m/s t =? = 0 (Remembe, at maximum height, the etical component of the elocity = 0) = u + at 0 = ( 10) t t = 5 s (b) u = +50 m / s a = 10 m / s d =? = 0 = u + ad 0 = 50 d = 15 m + ( 10)d This is the height aboe the stating point. To find the maximum height aboe the gound, add 10 m to obtain 45 m. The School Fo Excellence 016 The Essentials Physics Book 1 Page 19

6 Altenatie Solution Total enegy at maximum height = Total enegy at stating point 1 1 m + mgh = mu + max mgh initial gh = u + max gh initial 1 1 (86.6) + (10) h = (100) + (10)(10) Note: at max height = 86.6 m/s max (10) h max = hmax = 45 m (c) u = +50 m / s a = 10 m / s t =? d = 10 m d = ut + 1 at 10 = 50t + t 1 ) ( = 100t 10t 4 = 10t t t 10t 4 = 0 ( t 1)( t + ) = 0 t = 1 o t = We eject the negatie answe. The time of flight is theefoe 1 s. (d) d = t d 86.6 = 1 d = 1040 m The School Fo Excellence 016 The Essentials Physics Book 1 Page 0

7 (e) On impact, the ball s elocity will hae both a hoizontal and a etical component. The hoizontal component is, of couse, 86.6 m/s. We need to find the etical component. u = +50 m/ s a = 10 m / s =? d = 10 m = u + ad = 50 + ( 10)( 10) = = 4900 = ± 4900 = ±70 Since it is moing downwads at impact, the appopiate answe is = 70 m / s. The final elocity is the sum of the etical and hoizontal components m/s θ 70 m/s So, = 111m / s at an angle of 39 below the hoizontal. The School Fo Excellence 016 The Essentials Physics Book 1 Page 1

8 QUESTION 10 At a ca show a die plans to die up a 0 o amp and jump he ca acoss a ow of paked cas: She plans to be taelling at 30 ms -1 at the instant the ca leaes the amp. The landing amp on the opposite side of the gap is at the same height as the launching amp. Calculate the maximum gap that the ca can safely jump. (Model the ca as a point object.) Solution The School Fo Excellence 016 The Essentials Physics Book 1 Page

9 AIR RESISTANCE Ai esistance is a nuisance (unless you ae a paachutist). Being a fictional foce, it is always in the opposite diection to the elocity of the object. In magnitude, it changes accoding to the speed of the object (it being geate at highe speeds). Since a pojectile is always changing its elocity, the ai esistance is also always changing, in both magnitude and diection. This means that pojectile motion calculations that include ai esistance ae beyond the abilities of mee seconday school students. You ae, howee, equied to undestand the qualitatie effects of ai esistance. These can be summaised as follows: Ai esistance always opposes the motion of the pojectile. The magnitude of the ai esistance inceases as the speed of the pojectile inceases. The pojectile tansfes some of its kinetic enegy to the ai in the fom of tubulence. Fo an object pojected fom gound leel, the path would be as follows: path without ai esistance path with ai esistance With ai esistance, the following diffeences ae appaent: The aesthetically pleasing symmety no longe exists. The maximum height is eached beyond the mid-hoizontal position. The angle of impact is geate that the angle of pojection. Reduced hoizontal displacement. Reduced etical displacement. The final speed of the pojectile will be less than the initial speed (due to tansfe of enegy to the ai). The School Fo Excellence 016 The Essentials Physics Book 1 Page 3

10 CENTRIPETAL (CIRCULAR) MOTION CIRCULAR MOTION AT CONSTANT SPEED When an object moes along some path, its elocity ecto at any gien point is always a tangent to the path at that point. In the following diagam, the elocity ecto fo an object at point P is shown. As the object moes along the path, fom point P to point Q and point R, etc., its elocity ecto changes diection. Velocity P Path of object Q R Cente The changing elocity implies that a foce is acting in ode to bing about the change in elocity. An object can moe along a cicula path only if an extenal net foce causes it to do so if thee wee no net foce, it would moe in a staight line. (Remembe Newton s Fist Law: An object will hae constant elocity unless it is acted upon by some net extenal foce). If the speed aound the cicle is constant, this implies that thee is nee a component of the extenal net foce acting in the diection of the elocity ecto, i.e. the foce must always be pependicula to the elocity. Since the elocity is always tangential to the cicle, the foce must be diected adially, i.e. towads the cente of the cicle. F F F cente The School Fo Excellence 016 The Essentials Physics Book 1 Page 4

11 The speed of the object can be found thus: distancecoeed speed = timetaken cicumfeence speed = peiod π = T Een when moing at constant speed, the object is actually acceleating because it is changing its elocity. The acceleation is gien by: a = The diection of the acceleation is the same as that of the net foce: Always towads the cente of the cicle. The net foce that keeps the object moing in a cicle is called the centipetal foce, F c. (The acceleation of the object is called centipetal acceleation, a c.) Velocity Path of object P Foce, acceleation Cente The magnitude of the foce can be found by substituting F c instead of F net. This gies: = into F net = ma and witing a c = m F c The School Fo Excellence 016 The Essentials Physics Book 1 Page 5

12 We can deie anothe set of fomulae thus: Substituting π = into T 4π =, we obtain: a c = T a c 4π 4π Substituting a c = into F ma net =, we obtain: F c = m T T EXAMPLE 9 A taffic enginee needs to put a maximum speed-limit sign on a dangeous bend. If a ca taels too fast the sideways fictional foce will not be lage enough to keep it on the oad. The maximum sideways fictional foce that the tye-oad combination can poduce on a dy day without slipping is estimated to be 3000 N fo a ca of mass 100 kg. If the bend is modelled on the ac of a cicle of adius m as shown in the diagam aboe, calculate the maximum speed that a ca can hae and emain on the oad without slipping. Solution The foce of fiction supplies the tuning foce: m 100 F FRICTION = 3000 = = 100 = 15.8 m / s The School Fo Excellence 016 The Essentials Physics Book 1 Page 6

4. Two and Three Dimensional Motion

4. Two and Three Dimensional Motion 4. Two and Thee Dimensional Motion 1 Descibe motion using position, displacement, elocity, and acceleation ectos Position ecto: ecto fom oigin to location of the object. = x i ˆ + y ˆ j + z k ˆ Displacement: