NYJC EJC Frames of Reference Notes

c 9814 H3 Physics Frames of Reference - Notes Page 1 H3 Topic 1 – Frames Of Reference (Non-Relativistic) Content • Reference frames • Inertial frames • Centre of mass frame Learning Outcomes Candidates should be able to: (a) state that a frame of reference is a set of coordinates that can be used to determine positions and times of events in that frame (b) show an understanding that Newton’s laws of motion are obeyed in all inertial frames of reference (c) recall and apply the Galilean transformation equations to solve problems relating observations in different inertial frames of reference (d) show an understanding that the centre of mass frame (or zero momentum frame) is the inertial frame in which the total linear momentum of the system is zero (e) solve one-dimensional collision problems by considering velocities relative to the centre of mass of the system (i.e. in the zero-momentum frame).

c 9814 H3 Physics Frames of Reference - Notes Page 2 1. The Concept of a Frame of Reference A frame of reference is a specific space or set of coordinates in which an observer makes measurements of physical quantities, such as the position and time of events. Every velocity or speed measurement is inherently relative to a "point of reference" taken to be stationary. For example, a car moving at is measured relative to the road's surface. If measured from a moving truck, that speed would differ. Generally, a coordinate system (often with an origin at the point of reference) is established to facilitate quantitative calculations. 1.1 Inertial Frames of Reference An inertial frame is one in which the point of reference is not undergoing acceleration. In these frames, Newton’s laws of motion are valid; specifically, an object with no net external force remains at rest or moves at a constant velocity. • The Earth is often treated as an approximate inertial frame, despite its small rotational and orbital accelerations. • Any frame moving at a constant velocity relative to an inertial frame is also an inertial frame. • Einstein’s first postulate suggests that all laws of physics remain the same in every inertial frame. 1.2 Non-Inertial Frames of Reference A non-inertial frame is one that is accelerating relative to an inertial frame. In such frames, Newton’s laws are not obeyed. For instance, an accelerating aeroplane is a non-inertial frame because it does not obey Newton's first law relative to a stationary observer.

c 9814 H3 Physics Frames of Reference - Notes Page 3 2. Galilean Transformation Equations 2.1 Coordinate Transformations Consider two inertial frames of reference S and S’ respectively. To keep things simple, we have omitted the z -axis. The reference frame S ’ is moving at a speed of u and the origin of the two reference frames coincide at time t = t’ = 0 such that their separation at a later time t is ut. The position of particle P can be described by coordinates x and y in reference frame S or coordinates x’ and y’ in reference frame S’. In a 3 -dimensional analysis, the position of particle P can be expressed by the Galilean Coordinate Transformation equations: ( )'x x ut= − 1 'yy= 'zz= 'tt =

c 9814 H3 Physics Frames of Reference - Notes Page 4 2.2 Velocity Transformations If particle P moves in the x -direction, its velocity vx as measured by an observer stationary in frame S is x dxv dt= and its velocity v’ x as measured by an observer stationary in frame S’ is ''x dxv dt= . Differentiating (1) with respect to time gives: 'dx dx udt dt= − We arrive at the Galilean Velocity Transformation equation: 'xxv vu= − where 'xv is the horizontal component of velocity with respect to reference frame S’, where xv is the horizontal component of velocity with respect to reference frame S, where u is the horizontal component of velocity of reference frame S’ with respect to S. Example 1 A boat crossing a river moves with a speed of 3.0 m s −1 relative to the water. The water in the river has a speed of 1.2 m s−1 due east relative to the Earth. (a) If the boat heads due north in the frame of reference of the wat er, determine the velocity of the boat relative to an observer standing on either bank. (b) If the boat travels with the same speed of 3.0 m s−1 relative to the river and is to travel due north as observed by a stationary on the bank, what direction should it head to? (c) In which direction, (a) or (b), does the boat head to cross the river in the shorter time?

c 9814 H3 Physics Frames of Reference - Notes Page 5 Solution Let vBR be the velocity of boat relative to river, vRE be the velocity of river relative to the Earth, vBE be the velocity of boat relative to the Earth. In all cases, vBE = vBR + vRE. [You can obtain this equation in the same way we did it for the car and truck above, namely starting from the triangle for the position vectors and tak ing the time derivatives. Do try it out if this equation does not come to you intuitively.]. When the boatman paddles due north relative to the water, its path is altered by the flow of the water. The velocity v BE as seen by a stationary observer on the bank of the river is shown in (a). For the boat to always move northward perpendicular to the bank as seen by the s tationary observer, the boatman must paddle at angle ϕ as shown in (b). (a) Velocity of boat relative to Earth −= + = += 22 22 1 BE BR RE 3.0 1.2 3.23 m sv vv in the direction θ −− = = = °    RE11 BR 1.2tan tan 21.8 3.0 v v (b) Velocity of boat relative to Earth −= − = −= 22 22 1 BE BR RE 3.0 1.2 2.75 m sv vv Direction of boat relative to river φ −− = = = °    RE11 BR 1.2sin sin 23.6 3.0 v v (c) Boat in (a) takes the shorter time to cross the river as it has the larger velocity component (3.0 m s−1) in the northward direction. φ vBE vRE vBR vRE vBEvBR (a) (b) vBR vRE vBE vRE vBR vBE (a) (b)

c 9814 H3 Physics Frames of Reference - Notes Page 6 Example 2 To a cyclist travelling due North along a straight road at 4.0 m s−1, the wind appears to come from the East. If he increases his speed to 9.0 m s−1, it appears to blow from the North-East. Determine the speed and direction of the wind. Solution Let vCE be the velocity of the cyclist relative to Earth, vWE be the velocity of the wind relative to the Earth, vWC be the velocity of the wind relative to the cyclist. It can be shown that vWC = vWE − vCE. [Try showing it!] The vector diagrams represent the two scenarios: From the two figures, we can deduce that −= 1 WC1 5.0 m sv . Therefore, −= + = += 22 22 1 WE WC1 CE 5.0 4.0 6.4 m svvv θ −− = = = °   CE11 WC1 4.0tan tan 38.75.0 v v vWE−vCE vWC1 vWE −vCE vWC2 −vCE vWC1 vWE −vCE vWC2 vWE

c 9814 H3 Physics Frames of Reference - Notes Page 7 Example 3 Ship A is travelling due North at 6.0 m s−1 and ship B, 10 km North-West of it initially, is travelling due East at 3.0 m s−1, as shown in the figure below. Determine (a) the shortest distance between the ships subsequently, and (b) the time taken for the ships to meet at this distance. Solution (a) We first determine the velocity vector of A relative to B. ( ) −=−+= 2 21 AB 3 6 45 m sv 1 3.0tan 26.66.0α − = = ° 45.0 18.4θα= −= ° Therefore, the shortest distance of approach is 10000 sin18.4° = 3162 m. (b) From the diagram, AC = 10000 cos18.4° = 9486.8 m So, time taken t = 9486.8 45 = 1414 s. 3.0 m s−1 10 km 6.0 m s−1 B A B A 10 km 3.0 m s−1 6.0 m s−1 B A C vAB rAB B C A vAB rAB

c 9814 H3 Physics Frames of Reference - Notes Page 8 3. Centre of Mass F