Relative Motion Inertial and Noninertial Frames of Reference Inertial Frames of Reference • A Frame of Reference is a coordinate system relative to which motion is described or observed • Inertial Frame of Reference occurs when that frame of reference moves at ZERO or CONSTANT VELOCITY The law of inertia holds in an inertial frame of reference Recall: Law of Inertia: An object at rest will stay at rest and and object in motion will stay in motion unless acted upon by an external force. Non-inertial Frames of Reference • Non-Inertial Frame of Reference is a frame of reference that ACCELERATES with respect to the inertial frame of reference The law of inertia does not hold in this frame of reference Relative Motion • Frame of Reference: A coordinate system relative to which motion is described or observed • Relative Velocity: the velocity of an object relative to a specific frame of reference • Usually, relative velocity is a velocity that is relative to a frame of reference that is itself moving with a velocity relative to another frame of reference. Relative Motion Notation: ๐ฃเดฑ = ๐ฃ๐๐๐๐๐๐ก๐ฆ ๏ First subscript = object ๏ Second subscript = frame of reference Example: An airplane (P) travels at 450 km/h [N], as measured from the ground/earth (E) ๐๐๐ธ = 450 ๐๐เตโ [๐] Example: • An airplane (P) travels at 100 km/h [S], as measured from the ground/earth (E) ๐ฝ๐ท๐ฌ = ๐๐๐ ๐๐เต๐ [๐บ] • The air is moving relative to the ground (earth) thus we also must consider the velocity of air relative to the earth: ๐ฝ๐จ๐ฌ • We also must consider the velocity of the plane relative to the motion of the air ๐ฝ๐ท๐จ • These 3 velocities are related through ๐ฝ๐ท๐ฌ = ๐ฝ๐ท๐จ + ๐ฝ๐จ๐ฌ • This relationship is true in 1D, as well as 2D and 3D (with the use of vector components) Terminology: In General: ๐๐จ๐ช = ๐๐จ๐ฉ + ๐๐ฉ๐ช ๐๐จ๐ซ = ๐๐จ๐ฉ + ๐๐ฉ๐ช + ๐๐ช๐ซ Relative Motion in 1-D • Add vectors algebraically using the relative velocity equation Relative Motion in 2-D at right angles • Add vectors using Pythagorean theorem to determine magnitude • Use inverse tangent ratio to determine direction Relative Motion in 2-D • Determine the x and y components of the resultant vector separately • Use Pythagorean theorem to determine the magnitude • Use inverse tangent ratio to determine direction • Could also use sine or cosine laws when possible Relative Motion • Any two-dimensional relative velocity problem is an application of vector addition. • There are two scenarios you must learn to identify, when reading a question. • Also, it is important to note that what happens in one direction is independent of what happens in the other and that only time of travel is common to both. (recall projectile motion) Example: • Passengers on a cruise ship are playing shuffleboard. The shuffleboard disc’s velocity relative to the ship is 4.2 m/s [F]. The ship is travelling in the same direction at 4.6km/h relative to the Earth where the water is stationary. a. What is the discs velocity relative to the Earth? b. Determine the disc’s velocity relative to the Earth when the disc is moving in the opposite direction as the ship. c. Determine the disc’s velocity relative to the Earth if there is a current in the water of 1.1 m/s [F]. Example: • A person decides to swim across a river 84 m wide that has a current moving with a velocity of 0.40 m/s [E]. The person swims at 0.7 m/s [N] relative to the water. a. What is the velocity of the person with respect to the shore. b. How long will it take to cross? c. How far downstream will the person land? d. In which direction should she swim if she lands at a point directly north of her starting position? Example: A plane is travelling over Hamilton with a velocity of 3.5 x 102 km/h [N 35° W] with respect to the air. The wind is moving at 62km/h [S]. ๏ Determine the velocity of the plane relative to the ground. ๏ Determine the displacement of the plane after 1.2 hours. Example • Suppose you are a pilot flying a small plane flying due South between northern Ontario and Barrie. You want to reach the airport in Barrie in 3.0 h. The airport is 450 km away, and the wind is blowing from the west at 50.0 km/h. Determine the heading and airspeed you should use to reach your destination on time. Relative Motion of two objects • For any two objects or reference frames, A and B, the velocity of A relative to B has the same magnitude, but opposite direction, as the velocity of B relative to A ๐๐จ๐ฉ = −๐๐ฉ๐จ Example: • A train is travelling 100 km/h relative to the Earth in a certain direction. Objects on Earth (ex. trees) appear to be moving at 100 km/h in the opposite direction as viewed from an observer who is on the train. Relative Motion Between Two Moving Objects https://www.youtube.com/watch?v=s1KX_BSIGL0\ https://www.youtube.com/watch?v=s1KX_BSIGL0\ To find the relative velocity of one moving object with respect to another moving object: ๐๐จ๐ฉ = ๐๐จ − ๐๐ฉ ๐๐ฉ๐จ = ๐๐ฉ − ๐๐จ * Make sure to take directions into account (ex. if objects are moving in same/opposite direction, this will affect the signs) Relative Motion Between Two Moving Objects Picture two trains moving North on parallel tracks: Train A travels at 30 km/h [N] and Train B travels at 40 km/h [N] • You, on train A, will observe a passenger on train B moving at 10km/h [N] (Vba = Vb – Va) – • The passenger on train B will observe you moving 10km/h [S] (Vab = Va – Vb) • If train B was moving [S] and train A was moving [N] on parallel tracks, to you on train A it would appear that train B is moving at 70 km/h [S] as it approaches you who are moving [N] (Vba= Vb-Va, taking direction into account)
