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Chapter 4 ⅗ Relative Motion & 2D Kinematics DEHS 2012-13 Physics 1 Frame of Reference • A frame of reference is a coordinate system in which to measure the position, velocity, acceleration, etc of objects in it • Two objects are considered to be in the same frame of reference if they are moving with the same velocity (same speed and direction) or rotating about the same axis • This course will only cover frames of reference that move with constant velocity (that do not accelerate or rotate) Notation for Relative Velocity • Take note of the notation system The velocity of object/frame a r v ab relative to object/frame b • Think of it as object/frame b as being at rest • Ex: a car traveling 25 mph north moves relative r to the ground v cg 25 mph North Generalized Relative Motion Eqns • Reversing the subscripts reverses the direction r r of the velocity: v ab v ba r r r • The general equation is v 13 v 12 v 23 where 1 & 3 are the objects and 2 can be anything Example 4.5-1 A person climbs up a ladder on a moving train with a velocity of 3 m/s relative to the train. If the train moves relative to the ground with a velocity of 5 m/s, what is the velocity of the person relative to an observer watching from the ground? Example 4.5-2 You are riding in a boat whose speed relative to the water is 6.1 m/s. The boat points at an angle of 25° upstream on a river flowing 1.4 m/s. (a) What is your velocity relative to the ground? (b) Suppose the speed of the boat relative to the water remains the same, but the direction in which it points is changed. What angle is required for the boat to go straight across the stream? (c) What is the boat’s speed relative to the ground be? Example 4.5-3 A couple is shopping on the 2nd floor of the mall. The wife wants to go to the 1st floor and the husband wants to the 3rd floor. The both hop on an escalator. The escalators moves them both along at 2 m/s. Both escalators make an angle of 45° with the horizontal. Find the husband’s velocity relative to his wife vhusband 45° 45° vwife Position & Displacement Vectors • The position vector is defined as: r r x xˆ y yˆ • The displacement vector is defined as: r r r r r f ri Average Velocity Vector r r • Since r is a vector, v av must r also be a vector as it is r times the scalar (1/Δt) – It is in the same direction r as r r – It will be shorter than r as long as t > 1 s r r r v av t Example 4.5-4 r A dragonfly is observed initially at the position r i 2.00 m xˆ 3.50 m yˆ r Three seconds later it is at the position r f 3 .00 m xˆ 5 .50 m yˆ What is the dragonfly’s displacement and average velocity during this time? Example 4.5-4 Find the speed and direction of motion for a rainbow trout whose r velocity is v 3.7 m/s xˆ 1.3 m/s yˆ Average Acceleration Vector r recall: a av r v t maybe more useful… r a av r r v f vi t Example 4-10 (1D acceleration) A car is traveling at 25 m/s in a direction 25° E of S when its driver hits the brakes. It comes to a stop 0.34 s later. a) Find the car’s change in velocity in unit vector notation and mag/dir notation. b) Find the car’s acceleration in unit vector notation and mag/dir notation. c) Find the car’s velocity 0.17 s after the driver hits the brakes. (in unit vector and mag/dir notation) Example 4-10 (1D acceleration) Instantaneous v and a vectors • If an object’s path is drawn, the velocity vector v is always in the direction of the object’s motion and will appear to be tangent to the object’s path • The acceleration vector a can point in directions other that the direction of motion, and in general it does – a is ALWAYS in the direction of Δv • The angle between the a and v vectors will determine if the speed and/or the direction of motion will change (summarized on the next slide) How the relative directions of v and a affect an object’s velocity Angle between v&a Effect on speed Effect on direction of motion θ = 0° (same direction) 0° < θ < 90° speed up stays same speed up turns toward a θ = 90° remains constant turns toward a 90° < θ < 180° slows down turns toward a θ = 180° (opposite direction) slows down stays the same Example 4-10 (2D acceleration) Consider a car traveling in a circular path at a constant speed of 12 m/s. The car if it takes 10 s to complete ¼ of a revolution. Calculate the car’s average acceleration (in unit vector and mag/dir notation) for the portion of r r r f that 7 6.4 r ithe 76car’s .4 m motion yˆ is m a¼ xˆ revolution from to Example 4-10 (2D acceleration) What is the effect on the object’s speed and direction of motion at points 1, 2, 3, and 4? 2D Kinematic Equations • We can recycle the 1D kinematic equations, we just have to be sure to separate all vectors (r, vi, vf, a) into their components • Our overall technique will be to TREAT EACH DIRECTION INDEPENENTLY! x v ix t 1 2 ax t 2 y v iy t 1 2 ay t v x v ix a x t v y v iy a y t v v v v 2 x 2 ix 2 axx 2 y 2 iy 2 ayx 2 2D Motion w/ Constant Velocity Consider a turtle moving over a surface. The turtle moves with a speed of 5.0 m/s in a direction of 25° above the +x axis. How far has the turtle moved in the x and y directions after 5 sec? Example 4-12 A boat will travel at a speed of 6.5 m/s in still water. It aims directly across the river. The river’s current is 4.5 m/s west. a) What is the boat’s velocity (in mag/dir notation) relative to the shore? b) How long did it take the boat to cross the river the river is 1.25 km wide? c) How far downstream is the boat when it reaches the other side? d) In what direction should the boat aim so that it heads directly across the river? e) What would be the boat’s speed while traveling directly across the river? Example 4-13 A plane whose airspeed is 200 km/h heads due north toward its destination. But a 100 km/h northeast wind (that is, coming from the NE, pointing toward SW) suddenly begins to blow. a) What the the resulting velocity of the plane with respect to the ground? c) What direction should the pilot fly the plane to reach his destination? (d) How long would it take the pilot, assuming he corrected his course, to reach his destination? Example #-# A ball is rolled across a flat table with an initial velocity of 3.0 m/s North. A fan blows air to the east that causes the ball to accelerate at 0.4 m/s2 in that direction. a.) Find the position of the ball after 4.5 sec. b.) Find the speed of the ball after 10 sec. Example #-# A skier is accelerating down a 30° hill at 3.80 m/s2. Assume she starts from rest and accelerates uniformly. Her elevation change is 250 m. a.) What was the skier’s total displacement? b.) How long will it take her to reach the bottom of the hill? c.) What is the skier’s velocity at the bottom of the hill? (expressed in unit vector notation) d.) What is the skier’s speed at the bottom of the hill?