1. Kinematics in Two Dimensions
Chapter 3
Kinematics in Two Dimensions

2. Position
The position of an object is described by its position vector,

3. Displacement
The displacement of the object is defined as the change in its position,

4. Velocity
Average velocity
Instantaneous velocity

5. Instantaneous velocity
Vector of instantaneous velocity is always tangential to the object’s path at the object’s position

6. Acceleration
Average acceleration
Instantaneous acceleration

7. Acceleration Acceleration – the rate of change of velocity (vector)
The magnitude of the velocity (the speed) can change – tangential acceleration
The direction of the velocity can change – radial acceleration
Both the magnitude and the direction can change

8. Projectile motion A special case of 2D motion
An object moves in the presence of Earth’s gravity
We neglect the air friction and the rotation of the Earth
As a result, the object moves in a vertical plane and follows a parabolic path
The x and y directions of motion are treated independently

9. Projectile motion – X direction
A uniform motion: ax = 0
Initial velocity is
Displacement in the x direction is described as

10. Projectile motion – Y direction
Motion with a constant acceleration: ay = – g
Initial velocity is
Therefore
Displacement in the y direction is described as

11. Projectile motion: putting X and Y together

12. Projectile motion: trajectory and range

13. Projectile motion: trajectory and range

14. Chapter 3
Problem 47
The drawing shows an exaggerated view of a rifle that has been “sighted in” for a 91.4-meter target. If the muzzle speed of the bullet is v0 = 427 m/s, what are the two possible angles q1 and q2 between the rifle barrel and the horizontal such that the bullet will hit the target? One of these angles is so large that it is never used in target shooting.

15. Relative motion
Reference frame: physical object and a coordinate system attached to it
Reference frames can move relative to each other
We can measure displacements, velocities, accelerations, etc. separately in different reference frames

16. Relative motion
If reference frames A and B move relative to each other with a constant velocity
Then
Acceleration measured in both reference frames will be the same

17. Questions?