Relative Motion Vector Addition and Subtraction Motion in

-Relative Motion -Vector Addition and Subtraction -Motion in Two Dimensions Intro Physics Mr. Berman

Part I n n Relative Velocity Vector Addition and Subtraction (Graphical)

Relative Velocity of A relative to B: VAB=VA-VB v. AB : v of A with respect to B v. B : v of B with respect to a reference frame (ex. : the ground) v. A : v of A with respect to a reference frame (ex. : the ground)

Example 1 n n The white speed boat has a velocity of 30 km/h, N, and the yellow boat a velocity of 25 km/h, N, both with respect to the ground. What is the relative velocity of the white boat with respect to the yellow boat? Answer: 5 km/h, N

Example 2 The Bus Ride A passenger is seated on a bus that is traveling with a velocity of 5 m/s, North. If the passenger remains in her seat, what is her velocity: a) with respect to the ground? b) with respect to the bus?

Example 2 -continued The passenger decides to approach the driver with a velocity of 1 m/s, N, with respect to the bus, while the bus is moving at 5 m/s, N. What is the velocity of the passenger with respect to the ground? Answer: 6 m/s, N

Resultant Velocity The resultant velocity is the net velocity of an object with respect to a reference frame.

Example 3 - Airplane and Wind An airplane has a velocity of 40 m/s, N, in still air. It is facing a headwind of 5 m/s with respect to the ground. What is the resultant velocity of the airplane?

What if you have motion in two dimensions?

Motion in Two Dimensions n n Constant velocity in each of two dimensions (example: boat & river, plane and wind) Projectiles (constant velocity in one dimension and constant acceleration in the other dimension)

Graphical Addition of Vectors n Head-to-Tail Method n Parallelogram Method

Some rules to use for vector addition: n n Vectors can be moved parallel to themselves. Their magnitude and direction is still the same. The order of vector addition does not effect the resultant (commutative property).

Head-to-Tail Method of Method Addition n n Move one vector parallel to itself, so that its head is adjacent to the tail of the other vector. Draw the resultant by starting at the first tail and ending at the last head. A B Resultant

Add vectors A+B using the head to tail method: A Resultant B

Parallelogram Method of Vector Addition Place the vectors tail to tail forming a parallelogram. n Draw the diagonal from the two tails. This is the resultant. n A Resultant B

Note n n If the drawing is done to scale, measure the resultant. Convert the value of the resultant using the scale of the drawing.

Add vectors A+B using the parallelogram method: B A B Resultant

Add the following vectors using the head-to-tail method: Resultant

Graphical Vector Subtraction When subtracting A-B : n n Invert vector B to get -B Add A+(-B) normally

Subtract vectors A-B graphically: Resultant A B -B

Part II n Constant velocity in each of two dimensions (example: boat & river, plane and wind) Velocity of River with respect to the ground Velocity of Boat in Still Water

Adding vectors that are at 900 to each other. n n n Draw the vector diagram and draw the resultant. Use the Pythagorean Theorem to calculate the resultant. Use θ=tan-1(y/x) to find the angle between the horizontal and the resultant, to give the direction of the resultant. (00 is along the +x axis)

Example 4 -Airplane and Wind An airplane is traveling with a velocity of 50 m/s, E with respect to the wind. The wind is blowing with a velocity of 10 m/s, S. Find the resultant velocity of the plane with respect to the ground. Answer: 51 m/s, at 11 o below the + x axis (E).

Independence of Vector Quantities n Perpendicular vector quantities are independent of one another.

Independence of Vector Quantites n Example: The constant velocities in each of the two dimensions of the boat & river problem, are independent of each other. Velocity of River with respect to the ground Velocity of Boat in Still Water

Example 5 - Boat and River a) b) A boat has a velocity of 4 m/s, E, in still water. It is in a river of width 150 m, that has a water velocity of 3 m/s, N. What is the resultant velocity of the boat relative to the shore. How far downstream did the boat travel? Answer: a) 5 m/s, @ 37 o above + x axis (E) b) 113 m