Physics 110 Spring D Motion Problems: Projectile Motion Their Solutions

Transcription

1 Physcs 110 Sprn 006 -D Moton Problems: Projectle Moton Ther Solutons 1. A place-kcker must kck a football from a pont 36 m (about 40 yards) from the oal, and half the crowd hopes the ball wll clear the crossbar, whch s 3.1m hh. When kcked the all leaes the round wth a speed of 0 m/s at an anle of 53 0 to the horzontal. a. Does the ball clear or fall short of the crossbar? b. Does the ball approach the crossbar whle stll rsn or whle falln? (a) From our equatons of moton, the horzontal elocty s constant. Ths es us the flht tme for any horzontal dstance startn wth ntal x elocty cosθ. Thus the ertcal heht of x the trajectory s en as y x tan θ. Wth x 36.0 m, 0.0 m/s, and θ cos θ (9.80 m/s 53.0, we fnd y (36.0 m)(tan 53.0 ) )(36.0 m) ()(0.0 m/s) cos 3.94 m. The ball clears the 53.0 bar by ( ) m m. (b) The tme the ball takes to reach the maxmum heht s t 1 sn q (0.0 m/s)(sn 53.0 ) 9.80 m/s 1.63 s. The tme to trael 36.0 m horzontally s t x, whch es t x 36.0 m (0.0 m/s)(cos 53.0 ).99 s. Snce t > t 1 the ball clears the oal on ts way down.. A ball s tossed from an upper-story wndow of a buldn. The ball s en an ntal elocty of 8m/s at an anle of 0 0 below the horzontal, where t strkes the round 3 seconds later. a. How far horzontally from the base of the buldn does the ball strke the round? b. At what heht was the ball thrown? c. How lon does t take the ball to reach a pont 10m below the leel of launchn? (a) x x t (8.00 cos 0.0 )(3.00).6 m (b) Takn y poste downwards, y y t + 1 t 8.00(cos 0.0 ) (9.80)(3.00) 5.3 m

2 (c) cos 0.0 t + 1 (9.80) t, whch es a quadratc n t. The solutons to 4.90t +.74t are en by the quadratc formula, t.74 ± (.74) s 3. A frefhter 50m away from a burnn buldn drects a stream of water from a fre hose at an anle of 30 0 aboe the horzontal as shown below. If the speed of the stream s 40m/s, at what heht wll the water strke the buldn? From our equatons of moton, the horzontal elocty s constant. Ths es us the flht tme for any horzontal dstance startn wth ntal x elocty cosθ. x Thus the ertcal heht of the trajectory s en as y x tan θ. cos θ Substtutn the known quanttes, we fnd h 18.7m. 4. As some molten metal splashes off of the round, one droplet fles off to the east wth an ntal speed at an anle θ aboe the horzontal whle the other drop fles off to the west wth the same speed and at the same anle aboe the horzontal. In terms of and θ, fnd the dstance between the droplets as a functon of tme. At any tme t, the two drops hae dentcal y-coordnates. The dstance between the two drops s then just twce the mantude of the horzontal dsplacement ether drop has underone. Therefore, d x(t) ( x t) ( cos θ )t cosθ. t

3 5. A projectle s fred up an nclne (nclne anle φ) wth an ntal speed at an anle θ wth respect to the horzontal (θ > φ) as shown below. a. Show that the projectle traels a dstance d up the nclne, where d s cosθ sn( θ φ) en as d. cos φ (a) y tan(θ ) x cos (θ ) x. Settn x dcos(φ), and y dsn(φ), we hae dsn(φ) tan (θ )dcos(φ) cos (θ ) (dcos(φ)). Soln for d yelds, d cos (θ ) [sn (θ ) cos (φ) sn (φ) cos (θ )] cos (φ) or d cos (θ ) sn (θ φ) cos (φ). 6. A student decdes to measure the muzzle elocty of the pellets from hs BB un, whch s ponted horzontally. The shots ht the taret a ertcal dstance y below the un. a. Show that the ertcal dsplacement component of the pellets when traeln throuh the ar s en by y Ax, where A s a constant. b. Express the constant A n terms of the ntal elocty of the projectles and the acceleraton due to raty. c. If x 3.000m, and y 0.10m, what s the ntal speed of the pellets? (a)(b) Snce the shot leaes the un horzontally, the tme t takes to reach the taret s t x. The ertcal dstance traeled n ths tme s y 1 t x Ax where A (c) If x 3.00 m, y 0.10 m, then A A m/s 14.5 m/s.33 x 10, so that

4 7. A basketball player who s m tall s standn on the floor 10m from the basket. If the ball s shot at a 40 o anle wth the horzontal, at what ntal speed must the ball be thrown so t oes throuh the hoop wthout strkn the backboard? Assume that the basket heht s 3.05m 10.0 m x x t t cos 40.0 Thus, when x 10.0 m, t cos At ths tme, y should be 3.05 m.00 m 1.05 m. Thus, 1.05 m ( sn 40.0 ) 10.0 m cos (9.80m / s ) 10.0m cos 40.0 o. From ths, 10.7 m/s 8. When baseball players throw the ball n from the outfeld, they usually allow t to take one bounce before t reaches the nfelder on the theory that the ball arres sooner. Suppose that the anle at whch the bounced ball leaes the round s the same as the anle at whch the outfelder launched t but that the ball s speed after the bounce s one half of what t was before the bounce. a. Assumn the ball s thrown at the same ntal speed, at what anle θ, should the ball be thrown n order to o the same dstance D wth one bounce (blue path) as the ball thrown upward at 45 o wth no bounce (reen path)? b. Determne the rato of the tmes for the one-bounce and no bounce throws. a. Assumn that the projectle starts and ends at the same heht (whch s not strctly true) we can use the equatons for the rane of the projectle ben thrown at 45 o, the reen cure, to fnd R45 sn 90. For the ball that bounces one tme, the blue cure we

5 hae Rbounce R1 + R sn θ + sn θ. Here we requre the two equatons for the rane to be equal, and ths determnes the anle q for the blue path. Thus we hae, 4 o R 45 Rbounce sn 90 sn θ + sn θ sn θ θ b. The tme for any symmetrc parabolc flht s en as 1 y y + ( snθ ) t 1 t 0 (( sn θ ) t)t where the soluton t 0 s f snθ the tme the ball s thrown and the tme t s the tme to land. At 45 o sn 45, we fnd the landn tme to be t45 and for the ball bouncn at 6.6 o, the total landn tme s en as sn 6.6 sn sn 6.6 ttotal bouncn t1 + t +. Thus the rato of the tme for bouncn to the tme for no bounce s ttotal bouncn 3 sn Ths es the tme for bouncn t sn no bounce ery slhtly less than the tme wth no bouncn. 9. A Northrop B- Stealth bomber s flyn horzontally oer leel round, wth a speed of 75m/s at an alttude of 3000m. Nelect ar resstance n the follown problems. a. How far wll a bomb trael horzontally between ts release and ts mpact on the round? b. If the plane mantans ts ornal course and speed, where wll t be when the bomb hts the round? c. At what anle from the ertcal should the bombsht be set so that the bomb wll ht the taret seen n the sht at the tme of release? 6.jp

6 (a) y 1 t ; x t. Combne the equatons elmnatn t: y 1 ths ( x) 6.80 km y. Thus x y 75 ( 300) 9.80 x m from (b) (c) The plane has the same elocty as the bomb n the x drecton. Therefore, the plane wll be 3000 m drectly aboe the bomb when t hts the round. When θ s measured from the ertcal, tan θ x y ; therefore, x θ tan 1 y tan A person standn at the top of a hemsphercal rock of radus R kcks a ball (ntally at rest) to e t a horzontal elocty. a. What must be the rock s mnmum speed f the ball s neer to ht the rock after t s kcked? b. Wth ths ntal speed, how far from the base of the rock does the ball ht the round? Measure hehts aboe the leel round. The eleaton y b of the ball follows y b R t wth x t so y b R x / (a) The eleaton y r of ponts on the rock s descrbed by y r + x R. We wll hae y b y r at x 0, but for all other x we requre the ball to be aboe the rock surface as n y b > y r. Then y b + x > R Thus, R x + x > R, or R x R + x x > R and fnally, x x > x R. We et the strctest requrement for x approachn zero. If the ball's parabolc trajectory has lare enouh radus of curature at the start, the ball wll clear the whole rock: 1 > R, or when > R. (b) Wth R and y b 0, we hae 0 R x R or x from the rock's base s x R ( 1)R R. The dstance

7 11. An enemy shp s on the western sde of a mountan sland. The enemy shp can maneuer to wthn 500 m of the 1800 m hh mountan peak and can shoot projectles wth an ntal speed of 50m/s. If the eastern shore lne s horzontally 300m from the peak, what are the dstances from the eastern shore at whch a shop can be safe from the bombardment of the enemy shp? For ths problem, we need θ hh and θ low. From the daram, θ hh wll set the hhest eleaton that wll clear the mountan and en the closest rane, whle θ low wll set the lowest eleaton that wll clear the mountan and en the farthest rane. To fnd these anles we use our dsplacement equatons n the horzontal and ertcal drectons. Thus we hae x ( cosθ )t and y ( sn ) t 1 θ t. Soln the horzontal equaton of moton for t and substtutn ths nto the ertcal equaton of moton produces θ x 1 y x tan. Snce sn θ + cos θ 1, ddn by cos θ cos θ produces θ 1 tan + 1 Therefore we hae a quadratc n tan θ that wll cos θ determne θ hh and θ low. For the alues of x 500m and y 1800m, wth 50 m/s and 9.8m/s, we fnd tan θ {1.197, 3.906}. Usn the frst number we hae θ low 50.1 o and the second number θ hh 75.6 o. The rane at θ hh : m ( 50 ) sn( 151.) sn θ hh s 3 Rθ m. Thus the hh m 9.8 closest dstance to the shore s 3.07x10 3 m 300m 500m 70m from shore. sn m θhh ( 50 s ) sn( 100.) 3 The rane at θ low : Rθ m. Thus the low m 9.8 closest dstance to the shore s 6.8x10 3 m 300m 500m 3.48 x10 3 m from shore. s s So the safe dstances are less than 70m or reater than 3480m from shore.

8 1. The determned coyote s out once more to try to capture the eluse roadrunner. The coyote wears a par of Acme jet-powered roller skates, whch prode a constant horzontal acceleraton of 15 m/s. The coyote starts off at rest 70m from the ede of a clff at the nstant the roadrunner zps past hm n the drecton of the clff. a. If the roadrunner moes wth constant speed, determne the mnmum speed he must hae to reach the clff before the coyote. b. At the brnk of the clff the roadrunner escapes by makn a sudden turn, whle the coyote contnues straht off of the clff. If the clff s 100m aboe the floor of a canyon, where does the coyote land, assumn that hs skates reman horzontal and contnue to work whle n flht? c. What are the components of the coyote s mpact elocty? (a) Coyote: x 1 at ; (15.0) t : Roadrunner: x t; 70.0 t. Soln the aboe, we et.9 m/s and t 3.06 s (b) At the ede of the clff x at (15.0)(3.06) 45.8 m/s, y 1 a yt. Substtutn we fnd ( 9.80) t and x x t + 1 a xt (45.8)t + 1 (15.0) t. Soln the aboe es x 360 m t 4.5 s (c) For the Coyote's moton throuh the ar xf x + a x t (4.5) 114 m/s And yf y + a y t (4.5) 44.3 m/s