Projectile Motion. Physics 6A. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
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1 Projectile Motion Phic 6A
2 Projectile motion i a combination of Horizontal and Vertical motion We ue eparate et of formula for each, but both motion happen imultaneoul. Horizontal In the cae of projectile, with no air reitance, a = 0, o the formula become: v a 0 v 0 0 The horizontal component of V i contant! v 0 t Vertical The vertical component of the motion i jut free-fall. Thi mean the acceleration i contant, toward the ground, with magnitude g = 9.8m/ (1) () (3) Here are the formula: v v 0 v v 0 v 0 0 t gt 1 gt g 0
3 Let tr a couple of ample problem:
4 Sample Problem #1 A baeball i thrown with an initial peed of 30 m/, at an angle of 0 above the horizontal. When it leave the thrower hand the ball i.1 meter above the ground. In thi problem we will ignore air reitance. Another plaer want to catch the ball, alo at a height of.1 meter above ground. Where hould he tand?
5 Sample Problem #1 A baeball i thrown with an initial peed of 30 m/, at an angle of 0 above the horizontal. When it leave the thrower hand the ball i.1 meter above the ground. In thi problem we will ignore air reitance. Another plaer want to catch the ball, alo at a height of.1 meter above ground. Where hould he tand? Firt we need to et up a coordinate tem. A convenient wa to do it i to let the lowet point be =0, then call the upward direction poitive. With thi choice, our initial value are: 0 =.1m V 0 = 30 m/ v 0 = 30m/ 0 =.1m =0
6 Sample Problem #1 A baeball i thrown with an initial peed of 30 m/, at an angle of 0 above the horizontal. When it leave the thrower hand the ball i.1 meter above the ground. In thi problem we will ignore air reitance. Another plaer want to catch the ball, alo at a height of.1 meter above ground. Where hould he tand? Since thi i motion in dimenion, we will want to find the horizontal and vertical component of the initial velocit v 0 = (30m/)co(0 ) 8.m/ v 0 = (30m/)in(0 ) 10.3m/ V 0 = 30 m/ 0 =.1m 0 =0
7 Sample Problem #1 A baeball i thrown with an initial peed of 30 m/, at an angle of 0 above the horizontal. When it leave the thrower hand the ball i.1 meter above the ground. In thi problem we will ignore air reitance. Another plaer want to catch the ball, alo at a height of.1 meter above ground. Where hould he tand? Since thi i motion in dimenion, we will want to find the horizontal and vertical component of the initial velocit v 0 = (30m/)co(0 ) 8.m/ v 0 = (30m/)in(0 ) 10.3m/ V 0, 10.3m/ V 0, 8.m/ 0 =.1m =0
8 Sample Problem #1 A baeball i thrown with an initial peed of 30 m/, at an angle of 0 above the horizontal. When it leave the thrower hand the ball i.1 meter above the ground. In thi problem we will ignore air reitance. Another plaer want to catch the ball, alo at a height of.1 meter above ground. Where hould he tand? Now we need to figure out how far the ball will travel horizontall. The onl relevant formula we have for horizontal motion i 0 v 0 t We can ue 0 = 0, and we jut found v 0, But what hould we ue for t? V 0, 10.3m/ V 0, 8.m/ 0 =.1m =0
9 Sample Problem #1 A baeball i thrown with an initial peed of 30 m/, at an angle of 0 above the horizontal. When it leave the thrower hand the ball i.1 meter above the ground. In thi problem we will ignore air reitance. Another plaer want to catch the ball, alo at a height of.1 meter above ground. Where hould he tand? The time will be the ame a the time it take to travel up and then back down to a height of.1m. We need to ue the vertical motion to find thi. V 0, 10.3m/ V 0, 8.m/ 0 =.1m =0
10 Sample Problem #1 A baeball i thrown with an initial peed of 30 m/, at an angle of 0 above the horizontal. When it leave the thrower hand the ball i.1 meter above the ground. In thi problem we will ignore air reitance. Another plaer want to catch the ball, alo at a height of.1 meter above ground. Where hould he tand? The time will be the ame a the time it take to travel up and then back down to a height of.1m. We need to ue the vertical motion to find thi. There are a few option on how to proceed. In thi cae a imple and direct wa i jut to ue the baic vertical poition equation: v t gt V 0, 10.3m/ V 0, 8.m/ 0 =.1m =0
11 Sample Problem #1 A baeball i thrown with an initial peed of 30 m/, at an angle of 0 above the horizontal. When it leave the thrower hand the ball i.1 meter above the ground. In thi problem we will ignore air reitance. Another plaer want to catch the ball, alo at a height of.1 meter above ground. Where hould he tand? 4.9 m m 4.9 t.1 Here i the calculation: 1 0 v0,t gt.1m.1m t t 10.3 m 10.3 m t m t 0 m 10.3 t 0 t V 0, 10.3m/ V 0, 8.m/ 0 =.1m =0
12 Sample Problem #1 A baeball i thrown with an initial peed of 30 m/, at an angle of 0 above the horizontal. When it leave the thrower hand the ball i.1 meter above the ground. In thi problem we will ignore air reitance. Another plaer want to catch the ball, alo at a height of.1 meter above ground. Where hould he tand? Now that we have the time, we can ue our horizontal equation: v 8. m.1 m 0, t V 0, 10.3m/ V 0, 8.m/ 0 =.1m =0
13 Sample Problem # A bo 1.0m above the ground in a tree throw a ball for hi dog, who i tanding right below the tree and tart running at a contant peed the intant the ball i thrown. The bo throw the ball horizontall at 8.50m/, and air reitance can be ignored. (a) How fat mut the dog run to catch the ball jut a it reache the ground? (b) how far from the tree will the dog be when it catche the ball?
14 Sample Problem # A bo 1.0m above the ground in a tree throw a ball for hi dog, who i tanding right below the tree and tart running at a contant peed the intant the ball i thrown. The bo throw the ball horizontall at 8.50m/, and air reitance can be ignored. (a) How fat mut the dog run to catch the ball jut a it reache the ground? (b) how far from the tree will the dog be when it catche the ball? Here i a picture of the ituation. We can et up our coordinate tem with the origin at the bae of the tree. 1m V 0 V dog =?
15 Sample Problem # A bo 1.0m above the ground in a tree throw a ball for hi dog, who i tanding right below the tree and tart running at a contant peed the intant the ball i thrown. The bo throw the ball horizontall at 8.50m/, and air reitance can be ignored. (a) How fat mut the dog run to catch the ball jut a it reache the ground? (b) how far from the tree will the dog be when it catche the ball? Here i a picture of the ituation. We can et up our coordinate tem with the origin at the bae of the tree. Part (a) of thi problem i ea if we remember that the horizontal and vertical motion of the ball are independent. There i no acceleration in the -direction, o the ball will have a contant horizontal component of velocit. If the dog i going to catch the ball, hi horizontal velocit mut be the ame a the ball (the ball will be directl above the dog the whole time). So our anwer i 8.50 m/. 1m V 0 V dog = V 0
16 Sample Problem # A bo 1.0m above the ground in a tree throw a ball for hi dog, who i tanding right below the tree and tart running at a contant peed the intant the ball i thrown. The bo throw the ball horizontall at 8.50m/, and air reitance can be ignored. (a) How fat mut the dog run to catch the ball jut a it reache the ground? (b) how far from the tree will the dog be when it catche the ball? For part (b) we need to ue our free-fall formula to find out how long it take for the ball to get to the ground. Then we can ue the horizontal equation to find the ditance: (1) () (3) v v 0 v v 0 v 0 0 t gt 1 gt g 0 Equation (1) work becaue we know the initial and final height. 1m
17 Sample Problem # A bo 1.0m above the ground in a tree throw a ball for hi dog, who i tanding right below the tree and tart running at a contant peed the intant the ball i thrown. The bo throw the ball horizontall at 8.50m/, and air reitance can be ignored. (a) How fat mut the dog run to catch the ball jut a it reache the ground? (b) how far from the tree will the dog be when it catche the ball? For part (b) we need to ue our free-fall formula to find out how long it take for the ball to get to the ground. Then we can ue the horizontal equation to find the ditance: 0m 1m t m t m t 1m
18 Sample Problem # A bo 1.0m above the ground in a tree throw a ball for hi dog, who i tanding right below the tree and tart running at a contant peed the intant the ball i thrown. The bo throw the ball horizontall at 8.50m/, and air reitance can be ignored. (a) How fat mut the dog run to catch the ball jut a it reache the ground? (b) how far from the tree will the dog be when it catche the ball? For part (b) we need to ue our free-fall formula to find out how long it take for the ball to get to the ground. Then we can ue the horizontal equation to find the ditance: 0m 1m t m 0 v 0, 0 m t t 9.8 t 8.50 m m 1 m For an etra bonu challenge, tr thi problem again but have the bo throw the ball at an angle of 0 above the horizontal at peed 8.50 m/. Thi i worked out on the following lide, but pleae tr it on our own firt. 1m
19 Sample Problem # (bonu verion) A bo 1.0m above the ground in a tree throw a ball for hi dog, who i tanding right below the tree and tart running at a contant peed the intant the ball i thrown. The bo throw the ball at 8.50m/, at an angle of 0,and air reitance can be ignored. (a) How fat mut the dog run to catch the ball jut a it reache the ground? (b) how far from the tree will the dog be when it catche the ball? Here i a picture of the ituation. We et up our coordinate with the origin at the bae of the tree. 1m V dog =?
20 Sample Problem # (bonu verion) A bo 1.0m above the ground in a tree throw a ball for hi dog, who i tanding right below the tree and tart running at a contant peed the intant the ball i thrown. The bo throw the ball at 8.50m/, at an angle of 0,and air reitance can be ignored. (a) How fat mut the dog run to catch the ball jut a it reache the ground? (b) how far from the tree will the dog be when it catche the ball? Here i a picture of the ituation. We et up our coordinate with the origin at the bae of the tree. The firt thing we have to do i break the initial velocit into component: v 0 = (8.50m/)co(0 ) 7.99m/ v 0 = (8.50m/)in(0 ).91m/ Now part a) i jut like before: The dog mut run at 7.99 m/ to keep up with the ball. For part b) we can ue our -poition formula again, although thi time we might need the quadratic equation to olve it: 1m 0m 1m t v 0m 0,.91 m t m t 7.99 m m So the ball land 15.1 meter from the tree. t V dog =?
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