Motion in two dimensions (chapter three)

1. Motion in two dimensions (chapter three) • Kinematic quantities as vectors • Two dimensional motion with constant acceleration • Projectile motion • Uniform circular motion • Tangential and radial acceleration • Relative velocity

2. Position, velocity and acceleration vectors • Position: r(t) (bold indicates vector) • Velocity: v(t) • Acceleration: a(t)

3. Position, velocity and acceleration vectors • Exact same equations of kinematics can be used – now as vector equations • E.g. for a particle under constant acceleration: • (note “missing” equation – • division of vectors not yet • discussed

4. Position, velocity and acceleration vectors • What’s the point? • We won’t use the actual vector nature of these quantities much at first. • Realize that the entire motion can be described with vectors (simplified notation) • Motion is coupled through time variable • Circular motion – changes in both magnitude and direction

5. 2d motion with constant acceleration • Equivalent to two sets of independent motion – the parameter connecting them is time

6. Projectile motion • General assumptions • no air resistance • g is constant Initial velocity has components along x and y directions Acceleration has only y component = -g

7. Projectile motion This yields: The motion in the x and y directions are independent! Demos

8. Projectile motion • If we want a picture of the total motion of the object over time, we can solve the first equation for t and substitute it in the second • Valid for all xf, yf (dropped f subscript) • Equation of a parabola • Motion uniquely defined by vi and i • Note – use of tan only good for 0<i</2

9. Parabolic trajectory

10. Projectile motion – range equation(s)

11. Parabolic trajectory • Example: Derive the range equations in terms of the x and y components of the initial velocity • Peak time • Height • Range

12. Uniform circular motion • Direction of velocity changes  objects still accelerate • vector nature of acceleration • non constant acceleration – the change in x and y components of velocity is not constant

13. Uniform circular motion centripetal acceleration – constant acceleration along the direction towards the center of the circle Period of revolution

14. Radial and tangential acceleration • Velocity vector always tangent to path – acceleration in direction of motion is called tangential acceleration • The change in direction of the velocity is the component of acceleration along the direction of the instantaneous radius of curvature, called the radial acceleration

15. Radial and tangential acceleration • The total acceleration at any point of a curved path is the vector sum of the two. Note that the directions of the axes (radial and tangential) change as the object moves. • Example:What are the radial and tangential accelerations of an object undergoing projectile motion?

16. Relative velocity • rPO is the position of P measured in the O frame • rPO’ is the position of P measured in the O’ frame • vO’O is the velocity of the O’ frame measured in the O frame • Think about the motion – does the result make sense?

17. Relative velocity - examples 1. A boat’s speed in still water is 4.00 m/s. If the pilot wants to travel directly across a river flowing at 3.00 m/s, at what angle does he need to steer the boat? 2. If the same boat instead heads directly across the river, where will the boat end up if the river is 100. m wide? 3. Two cars approach an intersection at right angles and with the same speed (20.0 m/s). What is the relative velocity of one car with respect to the other?

18. Learning curve

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20. Summary (chapter two and three) • Kinematics – “how” of motion • Position, velocity and acceleration – vector nature, time derivatives • Average quantities – only depend on beginning and end points • Kinematic equations – constant acceleration • In two and three dimensions – motion along different axes is independent • Free fall, projectile motion • Motion along curved path – tangential and radial acceleration • Relative motion • Be careful about sig. figures, units