Physics 103: Lecture 5 2D Motion + Relative Velocities Today’s lecture will be on More on 2D motion Addition of velocities Velocity (like position and acceleration) is a vector. Recall a vector is a quantity with both direction and magnitude. A vector can be resolved into components (usually the axes of an orthogonal coordinate system) (NOT unique -- depends on the coordinate system. Addition of velocities is vector addition NOT algebraic addition. 1/2/2019 Physics 103, Spring 2004, U.Wisconsin 1

Summary of Lecture 4 Kinematics 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!) For kinematic problems this means one axis along the vector acceleration. Often this is gravity ie two axes are horizontal and vertical. x and y motions are independent! They share a common time t 1/2/2019 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin 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. 1/2/2019 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin Lecture 4, Pre-Flight 7&8 A flatbed railroad car is moving along a track at constant velocity. A passenger at the center of the car throws a ball straight up. Neglecting air resistance, where will the ball land? 1. Forward of the center of the car 2. At the center of the car 3. Backward of the center of the car correct 1/2/2019 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin 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 correct 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. since the monkey is going to start falling right away you need to aim below it Along the way, gravity is going to pull the dart down, so you would have to aim up. Aiming right at it or below would miss the monkey by going underneath it. 1/2/2019 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin Shooting the Monkey... x = v0 t y = -1/2 g t2 x = x0 y = -1/2 g t2 Dart hits the monkey! 1/2/2019 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin Shooting the Monkey... r = r0 - 1/2 g t2 At an angle, still aim at the monkey! Dart hits the monkey! r = v0 t - 1/2 g t2 1/2/2019 Physics 103, Spring 2004, U.Wisconsin

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 = v0 t - 1/2 g t2 Miss the enemy 1/2/2019 Physics 103, Spring 2004, U.Wisconsin

Reference Frames: Relative Motion VCB=VCA+VAB Velocity of B relative to ground ( C ) : VCB Velocity of A relative to ground ( C ) : VCA Velocity of B relative to A : VAB 1/2/2019 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin Relative Motion If an airplane flies in a jet stream, depending on the relative orientation of the airplane and the jet stream, the plane can go faster or slower than it normally would in the absence of the jet stream 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 Velocity is a vector - add velocities like vectors Sum the components Vx = V1x + V2x V1x = V1 cosq Vy = V1y + V2y V1x = V1 sinq 1/2/2019 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin 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 correct A B C 1/2/2019 Physics 103, Spring 2004, U.Wisconsin

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. A B C Time to get across = width of river/vertical component of velocity 1/2/2019 Physics 103, Spring 2004, U.Wisconsin

Lecture 5, Pre-Flight 3&4 (common misconceptions) While Carly is moving forward she will also be moving along with the current. two positive(+) direction motions = faster velocity. Carly will get there first because she is using the current to her advantage. A B C 1/2/2019 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin 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 correct vsg vwg vsw 1/2/2019 Physics 103, Spring 2004, U.Wisconsin

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 correct Seagull’s velocity in the reference frame of the wind = 9 m/s i. e., in this frame, 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. 1/2/2019 Physics 103, Spring 2004, U.Wisconsin

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 correct 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 1/2/2019 Physics 103, Spring 2004, U.Wisconsin

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 correct 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 1/2/2019 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin Once again: 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 correct 1/2/2019 Physics 103, Spring 2004, U.Wisconsin

Pendulum Motion - 2 Dimensions Kinematic equations for constant acceleration DO NOT APPLY for this case. 1/2/2019 Physics 103, Spring 2004, U.Wisconsin

Summary Relative Motion Velocity of seagull with respect to ground, Vsg Velocity of seagull with respect to wind, Vsw (i.e., velocity with which it would fly in calm winds) Velocity of wind with respect to the ground, Vwg 1/2/2019 Physics 103, Spring 2004, U.Wisconsin