-Relative Motion-Vector Addition and Subtraction

-Motion in Two Dimensions Intro

Physics Mrs.

Coyle

Part I

Relative Velocity Vector Addition and Subtraction

(Graphical)

Relative Velocity

Velocity of A relative to B: VAB=VA-VB

vAB : v of A with respect to B

vB : v of B with respect to a reference frame (ex.: the ground)vA : v of A with respect to areference frame (ex.: the ground)

Example 1

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

Answer: 5km/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 5m/s, N.

What is the velocity of the passenger with respect to the ground?

Answer: 6m/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 5m/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

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

Head-to-Tail Method

Parallelogram Method

Some rules to use for vector addition:

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

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.

B

Resultant

A

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

A B

Resultant

Place the vectors tail to tail forming a parallelogram.

Draw the diagonal from the two tails. This is the resultant.

A

B

Resultant

Parallelogram Method of Vector Addition

Note

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:

A

B

B

Resultant

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

Resultant

Graphical Vector Subtraction

When subtracting A-B :

Invert vector B to get -B Add A+(-B) normally

Subtract vectors A-B graphically:

AB

-B

Resultant

Part II

Constant velocity in each of two dimensions (example: boat & river, plane and wind)

Velocity of Boat in Still Water

Velocity of River

with respect to the ground

Adding vectors that are at 900 to each other. 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: 51m/s, at 11o below the + x axis (E).

Independence of Vector Quantities

Perpendicular vector quantities are independent of one another.

Independence of Vector Quantites Example: The constant velocities in each of

the two dimensions of the boat & river problem, are independent of each other.

Velocity of Boat in Still Water

Velocity of River

with respect to the ground

Example 5- Boat and River

A boat has a velocity of 4 m/s, E, in still water. It is in a river of width 150m, that has a water velocity of 3 m/s, N.

a) What is the resultant velocity of the boat relative to the shore.

b) How far downstream did the boat travel?

Answer: a) 5m/s, @ 37o above + x axis (E) b) 113m