2. Motion at Angles Life in 2-D

3. Review of 1-D Motion  There are three equations of motion for constant acceleration, each of which requires a different set of quantities.  Velocity and acceleration are vectors, since they have a magnitude and a direction. 1. 2. 3.

4. 1-D Vector Addition  If an object is moving within your frame of reference it will appear to be moving at its actual speed, but to an outsider it will appear to be moving faster or slower. How fast does the person on the train appear to be moving? v = Vector Addition Example

5. Adding Vectors In 2-D  1. Draw the first vector, beginning at the origin, with its tail at the origin.  2. Draw the second vector with its tail at the tip of the first vector.  3. Draw the Resultant (the answer) from the tail of the first vector to the tip of the last. Resultant

6.  Calculating the resultant’s magnitude and direction requires the use of right triangle mathematics.  We get the magnitude of the resultant from the Pythagorean Theorem.  c 2 = a 2 + b 2  R 2 = 5 2 + 2 2 = 25 + 4 = 29  R = (29) 1/2 = 5.4 meters

7.  We can get the direction of the resultant from the inverse tangent.  tan( θ ) = opp / adj = (2 meters) / (5 meters) = tan( θ ) = 0.40  θ = tan -1 (0.40) = 22 0  The Resultant is 5.4 meters in the direction of 22 0 North of East

8. Sample Problem  A ship leaves its home port expecting to go 500 km due south. Before it even moves 1 km, a severe storm blows the ship 100 km east. How far is the ship from its destination? In what direction must it travel to reach its destination?

9. Adding Vectors By Components  Any vector can be described as the sum as two other vectors called components. These components are chosen perpendicular to each other and can be found using trigonometric functions.

10. Sample Problem  If a ball is kicked with a velocity of 25 m/s at an angle of 37° above the horizontal, what is the horizontal and vertical component of its velocity?

11. Projectile Motion  A projectile is an object moving in two dimensions under the influence of Earth's gravity. Its path is a parabola. Projectile motion can be understood by analyzing the vertical and horizontal motion separately. The speed in the x-direction is constant while the speed in the y- direction is increasing.

12. Sample Problem  A car is found 25 m from the base of a 75 m high cliff. How fast was the car going when it went over the cliff?

13.  If an object is launched at an angle with the horizontal, the analysis is similar except that the initial velocity has a vertical component.

14. Sample Problem  A football is kicked with a velocity of 20 m/s at an angle of 60° above the horizontal. A) How long is it in the air for? B) What is the max height it reaches? C) How far does it travel?

15. Inclined Planes With Friction  We now have a second vector along the x axis. f k points in the negative direction (recall f k = μ k F N ).  The general solution for objects sliding down an incline is:

16. Sample Problem  A 5 kg block slides down a incline at an angle of 30 o with a coefficient of kinetic friction of.3 between the block and the incline.  a) Find the normal force on the block.  b) Find the force of friction between the block and incline.  c) Find the acceleration of the block down the incline.