Phys7: Lecture 4 Reminders All Discussion and Lab sections start meetin this week Homework is posted on course website Solutions to preious hwks will be posted Thursday mornins Today s Aenda 3-D Kinematics Independence of x and y components Baseball projectile Shoot the monkey Uniform circular motion 1 3-D Kinematics The position, elocity, and acceleration of a particle in 3 dimensions can be expressed as: r x i + y j + z k x i + y j + z k (i, j, k unit ectors ) a a x i + a y j + a z k We hae already seen the 1-D kinematics equations: x x(t) dx d d x a Pae 1
3-D Kinematics For 3-D, we simply apply the 1-D equations to each of the component equations. x x(t) dx x d x ax y y( t) dy y d y ay z z( t ) dz z d z az Which can be combined into the ector equations: r r(t) dr / a d r / 3 3-D Kinematics So for constant acceleration we can interate to et: a const + a t r r + t + 1 / a t (where a,,, r, r, are all ectors) Link to Actie Fiure 4-5 4 Pae
-D Kinematics Most 3-D problems can be reduced to -D problems when acceleration is constant: Choose y axis to be alon direction of acceleration Choose x axis to be alon the other direction of motion Example: Throwin a baseball (nelectin air resistance) Acceleration is constant (raity) Choose y axis up: a y - Choose x axis alon the round in the direction of the throw 5 x and y components of motion are independent. A man on a train tosses a ball straiht up in the air. View this from two reference frames: Reference frame on the moin train. Reference frame on the round. 6 Pae 3
Problem: Mark McGwire clobbers a fastball toward center-field. The ball is hit 1 m (y o ) aboe the plate, and its initial elocity is 36.5 m/s ( ) at an anle of 3 o (θ) aboe horizontal. The center-field wall is 113 m (D) from the plate and is 3 m (h) hih. What time does the ball reach the fence? Does Mark et a home run? y θ h D 7 Problem... First let s examine the motion Link to Actie Fiure 4-7 Choose y axis up. Choose x axis alon the round in the direction of the hit. Choose the oriin (,) to be at the plate. Say that the ball is hit at t, x x Equations of motion are: x x y y -t x x t y y + y t - 1 / t 8 Pae 4
Problem... Use eometry to fiure out x and y : y y θ x y Find x cos θ. and y sin θ. x 9 Problem... The time to reach the wall is: t D / x (easy!) We hae an equation that tell us y(t) y + y t + a t / So, we re done...now we just plu in the numbers: Find: x 36.5 cos(3) m/s 31.6 m/s y 36.5 sin(3) m/s 18.5 m/s t (113 m) / (31.6 m/s) 3.58 s y(t) (1. m) + (18.5 m/s)(3.58 s) - (.5)(9.8 m/s )(3.58 s) (1. + 65.3-6.8) m 3.5 m Since the wall is 3 m hih, Mark ets the homer!! 1 Pae 5
Lecture 4, Act 1 Motion in D Two footballs are thrown from the same point on a flat field. Both are thrown at an anle of 3 o aboe the horizontal. Ball has twice the initial speed of ball 1. If ball 1 is cauht a distance D 1 from the thrower, how far away from the thrower D will the receier of ball be when he catches it? (a) D D 1 (b) D 4D 1 (c) D 8D 1 11 Lecture 4, Act 1 Solution The distance a ball will o is simply x (horizontal speed) x (time in air) x t To fiure out time in air, consider the equation for the heiht of the ball: When the ball is cauht, y y y y 1 + y t t 1 t t y t 1 t y two solutions t t y (time of catch) (time of throw) 1 Pae 6
x x t Lecture 4, Act 1 Solution So the time spent in the air is proportional to y : t Since the anles are the same, both y and x for ball are twice those of ball 1., y ball 1,1 ball y, y,1 x,1 x, Ball is in the air twice as lon as ball 1, but it also has twice the horizontal speed, so it will o 4 times as far!! 13 Trajectory of projectile (special case of no air resistance) Decompose into x and y motion x x + x t + ½ at ( cosθ)t No horizontal forces y y + y t + ½ at ( sinθ)t ½ t Eliminate t usin t x / ( cosθ) y x x cos θ ( tanθ ) Trajectory is a parabola Specified by only initial speed and launch anle 14 Pae 7
Maximum heiht of projectile (special case of symmetric trajectory) At peak y, so y y -t A sinθ t sinθ ta A h max + y t A ½ t A ( sinθ ) sinθ 1 sinθ sin θ 15 Horizontal rane of projectile (special case of symmetric trajectory) Total time in air is t A Rane R x total time in air ( θ ) R cos t A sinθ ( cosθ ) sin θ (recall sinθ sinθ cosθ ) Rmax at θ 45 Let s examine some trajectories Link to Actie Fiure 4-11 16 Pae 8
Shootin the Monkey (tranquilizer un) Where does the zookeeper aim if he wants to hit the monkey? ( He knows the monkey will let o as soon as he shoots! ) 17 Shootin the Monkey... If there were no raity, simply aim at the monkey r r r t 18 Pae 9
Shootin the Monkey... With raity, still aim at the monkey! r r - 1 / t r t - 1 / t Dart hits the monkey! 19 Recap: Shootin the monkey... x t y - 1 / t This may be easier to think about. It s exactly the same idea!! x x y - 1 / t Pae 1