Kinematics Relative Motion

Objectives Explain why an object never really has a “true velocity” Explain what an inertial reference frame is Explain what is meant by relative motion For an object moving in two dimensions, describe what is meant by the following statement: The x- and y-motions take place in simultaneous time, but they can be mathematically solved independently Using vector mathematics, solve relative velocity problems

Why is there NO true velocity? Right now you would probably describe your motion as stationary. Are you? The Earth is rotating, so could argue tht you are moving east at more than 400 m/s. But, that’s not all The Earth is also revolving around the sun Our solar system is rotating within the galaxy Our galaxy is also moving through space. I guess motion depends on how the problem is viewed, and where the coordinate axis is located and how it is oriented. Motion is always relative (never true). But, relative how and to what ?

Inertial reference frame A frame of reference that is either at rest or moving at constant velocity. Relative Motion Motions in an inertial reference frames are considered to be relative motions if The reference frames (coordinate axes) are oriented the same. The origins of the reference frames coincide at time t = 0 . All the motion is in the x-y plane. Essentially motion is relative to some other defined object, which anchors the coordinate axis system. The anchoring object needs to be in an inertial frame of reference (at rest or moving at constant velocity). The anchoring frame of reference can be regarded as stationary, even if it is moving, and then the motion of other objects can be measured relative to this reference frame using the coordinate axis system.

Relative Velocity Problems Relative velocity problems involve two or more objects moving relative to each other and with different velocities. Many of these problems involve diagonal motion. Mathematically independent: When an object moves diagonally its motion can be broken into x- and y-components. Mathematically the x-direction can be solved independently, without considering the motion in the y-direction, and vice versa. This means that the x- and y-directions are mathematically independent. Simultaneous time: The x- and y-directions do share one important variable. TIME. Even though we can solve the x- and y-directions independently the diagonal motion takes place in simultaneous time. The tools used to solve problems involving diagonal motion are vector components and vector addition. Both dimensions move at constant velocity: vx = x / t and vy = y / t

A boat has a top speed of 5. 0 m/s in still water A boat has a top speed of 5.0 m/s in still water. It travels 100 m downstream in a river that has a current of 3.0 m/s. Example 1 a. How fast does the boat appear to move relative to an observer on land? When two motions occur simultaneously the vector velocities can be added together. The boat is initially traveling downstream parallel to, and in the same direction as, the river’s current. The speed of the river current adds to the speed of the boat. Parallel vectors add like normal addition. b. How long does the boat take to travel 100 m downstream?

A boat has a top speed of 5. 0 m/s in still water A boat has a top speed of 5.0 m/s in still water. It travels 100 m downstream in a river that has a current of 3.0 m/s. Example 1 The boat turns around and travels upstream back to its starting point. c. How fast does the boat appear to move relative to an observer on land? The velocity of the boat and river are still parallel, but now the velocity of the river is opposite (negative) the boats motion. Subtract the speed of the river. d. How long does the boat take to travel 100 m back upstream? e. What was the round trip time (total time down and back) ?

A boat has a top speed of 4. 0 m/s A boat has a top speed of 4.0 m/s. The boat aims straight across a 50 m wide river that has a current of 3.0 m/s. Example 2 a. Sketch this scenario. b. How fast does the boat appear to move to an observer on shore? How long does it take for the boat to cross the river. IMPORTANT: Match horizontal distance with horizontal speed. vboat vriver v θ1 50 m θ2

Example 2 v vboat vriver Δy A boat has a top speed of 4.0 m/s. The boat aims straight across a 50 m wide river that has a current of 3.0 m/s. Example 2 d. How far downstream does the boat travel? Match vertical distance with vertical speed. The x- and y-motions happen simultaneous time. Use time in “c”. e. What angle, measured from a line across the river, does the boat follow? f. What angle, as measured from the shore, does the boat follow? vboat vriver v θ1 50 m θ2 Δy

A boat has a top speed of 4. 0 m/s A boat has a top speed of 4.0 m/s. The boat aims at an angle slightly upstream in order to arrive directly across a 50 m wide river with a current of 3.0 m/s. Example 3 a. Sketch this scenario. b. How fast does the boat appear to move to an observer on shore? c. How long does it take the boat to cross the river IMPORTANT: You must use a velocity that has the same direction as displacement. You the cross river displacement. Use cross river velocity. vboat vriver v θ1 50 m θ2

A boat has a top speed of 4. 0 m/s A boat has a top speed of 4.0 m/s. The boat aims at an angle slightly upstream in order to arrive directly across a 50 m wide river with a current of 3.0 m/s. Example 3 e. What angle, as measured from a line across the river, does the boat follow? f. What angle, as measured from the shore, does the boat follow? vboat vriver v θ1 50 m θ2