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Physics 103: Lecture 5 2D Motion + Relative Velocities

Physics 103: Lecture 5 2D Motion + Relative Velocities. Today’s lecture will be on More on 2D motion Addition of velocities Relative motion. Summary of Lecture 4 Kinematics in Two Dimensions. x = x 0 + v 0x t + 1/2 a x t 2 v x = v 0x + a x t v x 2 = v 0x 2 + 2a x  x.

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Physics 103: Lecture 5 2D Motion + Relative Velocities

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  1. Physics 103: Lecture 52D Motion + Relative Velocities • Today’s lecture will be on • More on 2D motion • Addition of velocities • Relative motion Physics 103, Fall 2010, UW- Madison

  2. Summary of Lecture 4Kinematics in Two Dimensions • x = x0 + v0xt + 1/2 axt2 • vx = v0x +axt • vx2 = v0x2 + 2axx • y = y0 + v0yt + 1/2 ayt2 • vy = v0y +ayt • vy2 = v0y2 + 2ayy Choose alignment of the coordinate system to simplify the problem. (Choice always allows simplification!) Physics 103, Fall 2010, UW- Madison

  3. Question? Without air resistance, an object dropped from a plane flying at constant speed in a straight line will 1. Quickly lag behind the plane. 2. Remain vertically under the plane. 3. Move ahead of the plane There is no acceleration along horizontal - object continues to travel at constant speed (same as that of the plane) along horizontal. Due to gravitational acceleration the object’s speed downwards increases. Physics 103, Fall 2010, UW- Madison

  4. A battleship simultaneously fires two shells at enemy ships from identical canons. If the shells follow the parabolic trajectories shown, which ship gets hit first? 1. Ship A. 2. Ship B. 3. Both at the same time Higher the shell flies, the longer it takes. What should the captain order if he wants to hit both ships at the same time? A B Physics 103, Fall 2010, UW- Madison

  5. Projectile Motion the shell Physics 103, Fall 2010, UW- Madison

  6. correct Lecture 5, Pre-Flight 1&2 You are a vet trying to shoot a tranquilizer dart into a monkey hanging from a branch in a distant tree. You know that the monkey is very nervous, and will let go of the branch and start to fall as soon as your gun goes off. On the other hand, you also know that the dart will not travel in a straight line, but rather in a parabolic path like any other projectile. In order to hit the monkey with the dart, where should you point the gun before shooting? 1 Right at the monkey 2 Below the monkey 3 Above the monkey If the shot is fired at the monkey the same time the monkey drops, both objects will fall at the same rate causing the shot to hit the monkey. Physics 103, Fall 2010, UW- Madison

  7. Dart hits the monkey! Shooting the Monkey... x= v0 t y= -1/2 g t2 x = x0 y= -1/2 g t2 No monkeys were harmed during the making of this slide Physics 103, Fall 2010, UW- Madison

  8. Dart hits the monkey! Shooting the Monkey... r= r0 - 1/2 g t2 • At an angle, still aim at the monkey! r= v0t - 1/2 g t2 Physics 103, Fall 2010, UW- Madison

  9. Shooting the Enemy Paratrooper... If a soldier wants to shoot down an enemy paratrooper descending at uniform speed, Se, where should he aim? • Above the enemy • At the enemy • Below the enemy • Answer depends on the enemy’s position and vertical speed. r= r0 - Se t r= v0t - 1/2 g t2 Miss the enemy Physics 103, Fall 2010, UW- Madison

  10. 2D motion of simple pendulum Physics 103, Fall 2010, UW- Madison

  11. Relative Motion The motion may have a different appearance as viewed from a different reference frame, but this can be explained by including the relative velocity of the reference frame in the description of the motion. If a person rows a boat across a rapidly flowing river and tries to head directly for the shore, the boat moves diagonally relative to the shore Physics 103, Fall 2010, UW- Madison

  12. Airplane in a wind The cross-country navigation of an aircraft involves the vector addition of relative velocities since the resultant ground speed is the vector sum of the airspeed and the wind velocity. The velocity of the plane with respect to the ground ,is equal to the velocity of the plane with respect to the air plus the velocity of the air with respect to the ground. Physics 103, Fall 2010, UW- Madison

  13. correct ABC Lecture 5, Pre-Flight 3&4 Three swimmers can swim equally fast relative to the water. They have a race to see who can swim across a river in the least time. Relative to the water, Beth (B) swims perpendicular to the flow, Ann (A) swims upstream, and Carly (C) swims downstream. Which swimmer wins the race? A) Ann B) Beth C) Carly Physics 103, Fall 2010, UW- Madison

  14. ABC Lecture 5, Pre-Flight 3&4(great answers) • Beth will reach the shore first because the vertical component of her velocity is greater than that of the other swimmers. • The key here is how fast the vector in the vertical direction is. "B" focuses all of its speed on the vertical vector, while the others divert some of their speed to the horizontal vectors. Time to get across = width of river/vertical component of velocity Physics 103, Fall 2010, UW- Madison

  15. correct Followup Question Heather wants to swim across a flowing river in such a way that she ends up on the opposite side directly opposite her starting point. She should therefore aim…. 1) upstream 2) downstream 3) directly across Physics 103, Fall 2010, UW- Madison

  16. correct Lecture 5, Pre-Flight 5 and 6 A seagull flies through the air with a velocity of 9 m/s if there were no wind. However, it is making the same effort and flying in a headwind. If it takes the bird 20 minutes to travel 6 km as measured on the earth, what is the velocity of the wind? 1. 4 m/s 2. -4 m/s 3. 13 m/s 4. -13 m/s • Seagull’s velocity relative to the wind = 9 m/s • i. e., in the frame relative to the wind, wind velocity is zero • Seagull travels at 6000/1200 = 5 m/s relative to earth. Therefore, the wind velocity relative to earth is 5-9=-4 m/s. Physics 103, Fall 2010, UW- Madison

  17. correct Follow-up, Pre-Flight 5 and 6 If the seagull turns around and flies back how long will it take to return? 1. More time than for flying out 2. Less time than for flying out 3. The same amount of time • Seagull’s return velocity is: -4-9=-13 m/s. The speed is higher so it takes less time to return. Time taken for the return is given by 6000 m / 13 (m/s) = 461.5 s = 461.5/60 = 7.69 minutes Physics 103, Fall 2010, UW- Madison

  18. correct Follow-up 2, Pre-Flight 5 and 6 How are the round-trip times with and without wind related if the seagull always goes at 9 m/s? 1. The round-trip time is the same with/without the wind 2. The round trip time is always larger with the wind 3. It is not possible to calculate this • Time taken for the round trip with wind is: 27.69 minutes • Time taken for the round trip without wind is: 12000 m / 9 m/s = 1333 s = 22.2 minutes Physics 103, Fall 2010, UW- Madison

  19. Once again: correct A boat is drifting in a river which has a current of 1 mph. The boat is a half mile upstream of a rock when an observer on the boat sees a seagull overhead. The observer sees the gull flying continuously back and forth at constant speed (10 mph) between the boat and the rock. When the boat passes close by the rock, how far (what distance) has the gull flown? • 5 miles • 10 miles • Not sufficient information to determine the distance Physics 103, Fall 2010, UW- Madison

  20. Summary Relative Motion Velocity of seagull with respect to ground, Vgs Velocity of seagull with respect to wind, Vws(i.e., velocity with which it would fly in calm atmosphere) Velocity of wind with respect to the ground, Vgw Physics 103, Fall 2010, UW- Madison

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