Ch 4: Motion in Two and Three Dimensions Fundamentals of Physics Halliday, Resnick, Walker AP Physics 8/24/2012

Introduction This chapter combines the concepts of 1-D motion and vectors to describe motion in two and three dimensions. What are some examples of 2-D and 3-D motion? AP Physics 8/24/2012

Agenda Position and Displacement Velocity (Average and Instantaneous) Acceleration (Average and Instantaneous) Projectile Motion Uniform Circular Motion Relative Motion in 1-D and 2-D AP Physics 8/24/2012

Vocabulary Projectile Motion Uniform Circular Motion Centripetal Acceleration Period of Revolution Relative Motion Reference Frame AP Physics 8/24/2012

Position Vector, r(t) The location of an object in space is specified by a position vector, r. The position, r(t), is a function of time. AP Physics 8/24/2012

Rabbit Example: Position r(t) A ‘wascally’ rabbit’s position is given by the parametric equations r(t) = xî + yĵ where x(t) = −0.31t2 + 7.2t + 28 y(t) = 0.22t2 − 9.1t + 30 Find r at t = 15 s. x(15) = 66m, y(15) = −57m |r| = 87 m, θ = −41o AP Physics 8/24/2012

Displacement Vector, Δr Displacement , Δr, of an object is the difference between two positions. Δr = r(t2) – r(t1) = r2 – r1 = (x2 − x1) î + (y2 − y1) ĵ = Δx î + Δy ĵ Displacement is not necessarily the same as the distance traveled. Why? AP Physics 8/24/2012

Rabbit Example: Plot r(t) Use parametric mode on graphing calculator to plot the rabbit’s position. Note: This is y vs x, not x vs t. AP Physics 8/24/2012

Rabbit Example: Displacement Δr Determine the rabbit’s displacement Δr over the interval t = 10s to t = 20s. r(10) = 69î − 39ĵ m r(20) = 48î − 64ĵ m Δr = r(20) − r(10) = (48 − 69) î + (−64 −(−39)) ĵ = −21 î −25 ĵ m |r| = 33 m @ θ = 230o ***Sketch the vectors*** AP Physics 8/24/2012

Average Velocity Vector, vavg vavg = displacement / time (same as 1-D case) vavg is in the same direction of Δr AP Physics 8/24/2012

Rabbit Example: vavg(t) Determine the rabbit’s average velocity over the interval t = 10 s to t = 20 s Determine vavg vavg = −2.1 î −2.5 ĵ m/s vavg = 3.3 m/s @ 230o AP Physics 8/24/2012

Instantaneous Velocity, v(t) v(t) is the velocity of the object at an instant in time. v(t) is the instantaneous rate of change of the object’s position, r, with respect to time, t. AP Physics 8/24/2012

Speed, |v(t)| Speed is the magnitude (or absolute value) of the instantaneous velocity. |v| = (vx2 + vy2 + vz2) ½ Speed is always ≥ 0. AP Physics 8/24/2012

Direction of the Instantaneous Velocity, v(t) The direction of v is always tangent to the object’s path at the object’s position. AP Physics 8/24/2012

Rabbit Example: Velocity, v v is tangent to the rabbit’s path at any instant. Determine v at t = 15 s. vx(t) = dx/dt = −0.62t + 7.2 vy(t) = dy/dt = 0.44t − 9.1 v (15) = −2.1î − 2.5ĵ m/s = 3.3 m/s @ −130o = 3.3 m/s @ 230o AP Physics 8/24/2012

Average Acceleration, aavg Average acceleration, aavg, is the ratio of the object’s change in velocity, Δv, and the corresponding time interval, Δt . AP Physics 8/24/2012

Instantaneous Acceleration, a(t) a(t) is the acceleration of the object at an instant in time. a(t) is the instantaneous rate of change of v with respect to t. AP Physics 8/24/2012

Rabbit Example: Acceleration, a Determine a at t = 15 seconds. ax(t) = dvx/dt = −0.62 m/s2 ay(t) = dvy/dt = 0.44 m/s2 a (15) = −0.62î − 0.44ĵ m/s2 = 0.76 m/s2 @ 145o AP Physics 8/24/2012

Speeding up or Slowing down? If an object is changing speed and / or changing direction, then it is accelerating. Given that θ is the angle between a and v when they are tail to tail, then If θ = 0o, the object is speeding up only. If θ < 90o, the object is speeding up and changing direction. If θ = 90o, the object’s speed is constant, but it’s changing direction. If θ > 90o, the object is slowing down and changing direction. If θ = 180o, the object is slowing down only. AP Physics 8/24/2012

Projectile Motion Special case of 2-D motion Horizontal motion: ax = 0 so vx = constant Vertical motion: ay = g = constant so the constant acceleration equations apply. Assumptions: Horizontal and vertical motions are independent of each other Air resistance (i.e., drag) can be ignored. AP Physics 8/24/2012

Projectile Motion Illustrated AP Physics 8/24/2012

Projectile Motion Equations Trajectory Equation: objects follow a parabolic path. Range Equation: gives range R = Δx only when Δy = 0. What angle gives maximum range? AP Physics 8/24/2012

Motion with Constant Acceleration v = vo + at x − xo = vot + ½ at2 v2 = vo2 + 2a(x − xo) x − xo = ½ (vo + v)t x − xo = vt − ½ at2 AP Physics 8/24/2012

Free-Fall Acceleration Equations If +y is vertically up, then the free-fall acceleration due to gravity near Earth’s surface is a = − g = − 9.8 m/s2. v = vo − gt y − yo = vot − ½ gt2 v2 = vo2 − 2g(y − yo) y − yo = ½ (vo + v)t y − yo = vt + ½ gt2 AP Physics 8/24/2012

Projectile Motion: Monkey-Hunter Demo The projectile and can fall at the same rate. The point of intersection depends on the launch velocity angle and speed. AP Physics 8/24/2012

Problem Solving Techniques Event 1 Event 2 Time, t x position x velocity x accel. Determine the two events that define the start and end of the motion. Identify the known and unknown quantities for the x and y motions. The x and y equations are independent, but are linked by time t. Event 1 Event 2 Time, t y position y velocity y accel. AP Physics 8/24/2012

Projectile Motion: Rescue Problem vo = 198 km/h, horizontal h = 500 m At what line of sight angle φ should the pilot release the raft to reach the swimmer? AP Physics 8/24/2012

Projectile Motion: Pirate Problem Range of pirate ship is R = 560 m. The cannon ball’s initial speed is vo = 82 m/s. At what angle(s) should the fort aim the cannon to hit the pirate ship? AP Physics 8/24/2012

Uniform Circular Motion Another special case of 2-D motion. An object travels in a circle of radius r at constant speed v. Centripetal (“center seeking”) acceleration Direction: perpendicular to v directed towards center of circle Magnitude: a = v2/r What causes centripetal acceleration? AP Physics 8/24/2012

Period of Revolution Period of revolution = Time required for the object to go around the circle once. AP Physics 8/24/2012

UCM: Top Gun Problem A Test Pilot flying a F-22 fighter aircraft at a speed of 2500 km/h initiates a horizontal turn of radius 5.80 km. What is the pilot’s centripetal acceleration? Given that loss of consciousness occurs at approximately 5g of acceleration, is the pilot in danger? Explain. AP Physics 8/24/2012

Relative Motion in 1-D An observer watches two joggers running by at 6 km/hr. What is the relative speed between the two joggers? Observers measure position, velocity, and acceleration relative to their frame of reference. The reference frame is the physical object to which we attach our coordinate system. Examples: ground, car, plane, ship AP Physics 8/24/2012

Relative Motion in 1-D Illustrated AP Physics 8/24/2012

Relative Motion Notation Observers in reference frames A and B measure the position of the same object P. xPA = xPB + xBA Position of Frame B relative to Frame A Position of P measured by observer in Frame A Position of P measured by observer in Frame B AP Physics 8/24/2012

Relative Motion Equations, 1-D AP Physics 8/24/2012

Relative Motion Equations, 2-D AP Physics 8/24/2012

Relative Motion in 2-D Illustrated AP Physics 8/24/2012

Frames of Reference Moving at Constant Relative Velocity Observers in different frames of reference may measure different positions and velocities of an object. The observers will measure the same acceleration if they move at constant velocity relative to each other. AP Physics 8/24/2012

1-D Frame of Reference Example An observer on a Train throws Ball in x-direction with velocity of vBT. Velocity of train relative to ground is vTG. An observer on the ground measures velocity of ball, vBG. vBG = vBT + vTG vBG km/h vBT vTG 50 100 −50 −25 25 AP Physics 8/24/2012

2-D Frame of Reference Example: Airplane Navigation Givens: Airspeed of plane is 215 km/h. Wind velocity is 65.0 km/h directed 20o East of North. The plane flies due East relative to the ground. Determine the plane’s heading and ground speed. AP Physics 8/24/2012

Summary 2-D and 3-D motion combines the concepts of 1-D motion and vectors. Know the various r, v, and a definitions. Special cases of 2-D motion: Projectile Motion Uniform Circular Motion Relative Motion and Frames of Reference. AP Physics 8/24/2012