Review from 2 nd Lecture

1. Review from 2nd Lecture • Kinematics of Constant Acceleration v = v0 + at x = x0 + v0t + at2/2 • Freely Falling Objects a = -g = 9.8 m/s2 • Vectors • Direction and Magnitude • Components • Unit Vectors • Addition & Subtraction

2. Kinematics in 2D (or more) • Position x goes to r • Displacement Dx goes to Dr = r2-r1 • Velocity v goes to v = Dr/Dt or dr/dt • Acceleration a goes to a = Dv/Dt or dv/dt • Generally replace scalars with respective vectors

3. Constant Acceleration in 2D • But now with (vector) a • Velocity Equation v = v0 + at • This is exactly vx = vx0 + axt vy = vy0 + ayt • Position Equation x = x0 + v0t + at2/2 • This is exactly x = x0 + vx0t + axt2/2 y = y0 + vy0t + ayt2/2

4. Projectile Motion • Projectile motion is a common example of using 2D kinematics with constant acceleration • Clue words: throw, fire, launch, jump, drop • The connecting theme is constant acceleration (from gravity) • We will always pick a coordinate system such that the acceleration is along the y-axis • Constant acceleration in y y = y0 + vy0t + ayt2/2 • No acceleration (constant velocity) in x x = x0 + vx0t

5. Projectile Motion • The Canonical Canon • Initial velocity vi, angle qi • Want to find height h and range R • Motion in x and y independent

6. Projectile Motion • While x and y have independent equations, still really vector equation

7. Projectile Motion • Find maximum height

8. Projectile Motion • Find range

9. Projectile Motion • Find y vs x

10. Relativity • Galilean Relativity, that is… • What happens when two observers moving at constant relative velocity make observations? • Observer A sees only vertical motion • Observer B sees a parabolic trajectory (projectile motion)

11. Galilean Relativity • As usual, one has to convert from one reference frame to another • Origin of S’ frame moves with velocity v0 with respect to frame S

12. Galilean Relativity • Note that acceleration does not change! • Since (as we’ll see) force is proportional to acceleration, (Newtonian) physics is the same in any two frames moving at constant relative velocity!

13. Uniform Circular Motion • Question: Does constant speed imply constant velocity? • No, vector can rotate with Dv perpendicular to v • This happens when an object moves in a circle at constant speed

14. Uniform Circular Motion • What is the magnitude of the acceleration?

15. Uniform Circular Motion Unit vectors which change direction! • Note that a is not constant, but its magnitude is • Can pick appropriate coordinate system and unit vectors for the problem • In polar coordinates r and dq/dt are constant • Break acceleration down into tangential and radial components • Tangential component is zero for uniform circular motion

16. Tangential & Radial Acceleration • Can use tangential & radial acceleration even when not on circle • Can also have non-zero tangential acceleration with circular motion • Rocket-on-a-string • Radial (centripetal) acceleration must increase in magnitude