1. Free Fall: Kinematics in two (or more) dimensions obeys the same 1- D equations in each component independently. Kinematics in two (or more) dimensions obeys the same 1- D equations in each component independently. Physics 1710 Chapter 4: 2-D Motion—II a x = 0 a y = -g v x =constant y = y initial – ½ gt 2 x = x initial + v x,initial t

2. Kinematic Equations Physics 1710 Chapter 4: 2-D Motion—II REVIEW x(t) = x initial + v initial t + 1/2 a o t 2 v(t) = dx/dt = v initial + a o t a= dv/dt = a o Observe Air Track

3. 1′ Lecture: The velocity and acceleration of a body in a moving (and accelerating) frame of reference (FoR ) is equal to that of a stationary FoR minus the velocity or acceleration of the moving FoR. The velocity and acceleration of a body in a moving (and accelerating) frame of reference (FoR ) is equal to that of a stationary FoR minus the velocity or acceleration of the moving FoR.  v ′ = v – v frame of reference  a ′ = a – a frame of reference Motion in a circle at a constant speed is due to an acceleration toward the center of the circle, a centripetal acceleration of Motion in a circle at a constant speed is due to an acceleration toward the center of the circle, a centripetal acceleration of  a = - r and |a| = v 2 / |r| toward the center.  a = - ω 2 r and |a| = v 2 / |r| toward the center. Physics 1710 Chapter 4: 2-D Motion—II

4. Frame of Reference Physics 1710 Chapter 4: 2-D Motion—II Fly v′v′v′v′ v′v′v′v′ v frame of reference v v = v′ + v frame of reference and v′ = v - v frame of reference

5. Relative Motion and the Galilean Transformation: and the Galilean Transformation: r ′ = r – v frame of reference t d r ′/dt = d r/dt – v frame of reference d r ′/dt = d r/dt – v frame of reference v ′ = v – v frame of reference d v ′/dt = d v/dt – d v frame of reference /dt d v ′/dt = d v/dt – d v frame of reference /dt a ′ = a – a frame of reference Physics 1710 Chapter 4: 2-D Motion—II

6. Relative Motion in a Free Falling Frame of Reference Physics 1710 Chapter 4: 2-D Motion—II In Lab Frame In Moving Frame a ′ = a – a frame of reference

7. Everyday Physics: When a bird is flying at the same velocity(same speed and direction) as a car what is its relative velocity v′ to the car? When a bird is flying at the same velocity(same speed and direction) as a car what is its relative velocity v′ to the car? When you brake an automobile, which direction is the acceleration on the vehicle? Which direction do the passengers sense as the acceleration relative to the frame of reference of the car?When you brake an automobile, which direction is the acceleration on the vehicle? Which direction do the passengers sense as the acceleration relative to the frame of reference of the car? Physics 1710 Chapter 4: 2-D Motion—II

8. Relative Motion and the Galilean Transformation: and the Galilean Transformation: v ′ = v – v frame of reference a ′ = a – a frame of reference Physics 1710 Chapter 4: 2-D Motion—II

9. Uniform Circular Motion θ r Physics 1710 Chapter 4: 2-D Motion—II x = R sin θ y = R cos θ R 2 = x 2 + y 2 r = (x,y) = x i + y j v v x = R cos θ d θ/dt = Rω cos θ v y = - R sin θ d θ/dt = - Rω sin θ d θ/dt = ω = constant

10. Uniform Circular Motion θ r Physics 1710 Chapter 4: 2-D Motion—II v v x = R cos θ d θ/dt = R ω cos θ v y = - R sin θ d θ/dt = - R ω sin θ a x = - R ω 2 sin θ a y = - R ω 2 cos θ a = - ω 2 r |a| = v 2 /R, toward center

11. Uniform Circular Motion—in review ⃒a⃒ = a, a constant value always pointing toward the center of the circle: Centripetal acceleration. a = a x i+ a y j = - ω 2 r where a x = a sin θ = - (v 2 /R) sin θ = - ω 2 R sin θ a y = a cos θ = - (v 2 /R) cos θ = - ω 2 R cos θ Physics 1710 Chapter 4: 2-D Motion—II

12. Uniform Circular Motion |a| = v 2 /R, toward center The Centripetal acceleration, where v is the tangential speed and R is the radius of the circle. v = ω R = 2πR / T, Where T is the “period” or time to make one revolution. |a |= 4π 2 R/ T 2 = ω 2 R Physics 1710 Chapter 4: 2-D Motion—II

13. Uniform Circular Motion Little Johnny on the Farm—part II. Physics 1710 Chapter 4: 2-D Motion—II - g a v a v. ?

14. Uniform Circular Motion - g a FoR Physics 1710 Chapter 4: 2-D Motion—II In Frame of Reference of Bucket a′ = -g – a FoR If |a FoR | ≽ |g| In same (down) direction, a′ is up!

15. Uniform Circular Motion g a Physics 1710 Chapter 4: 2-D Motion—II a = (2π/T) 2 R, T ≈1. sec R ≈1. m 2π ≈ 6. a ≈ (6./1.sec) 2 (1.m) a≈ 36. m/sec 2 a> 3 g Do you believe this?

16. 80/20 Summary: In a moving or accelerating Frame of Reference In a moving or accelerating Frame of Reference v ′ = v – v frame of reference v ′ = v – v frame of reference a ′ = a – a frame of reference a ′ = a – a frame of reference The Centripetal acceleration is The Centripetal acceleration is a = - r or |a| = v 2 / |r|, toward the center. a = - ω 2 r or |a| = v 2 / |r|, toward the center. Physics 1710 Chapter 4: 2-D Motion—II