Ground Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. Dr Tay Seng Chuan

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1 Ground Rules PC11 Fundamentals of Physics I Lectures 3 and 4 Motion in One Dimension Dr Tay Seng Chuan 1 Switch off your handphone and pager Switch off your laptop computer and keep it No talking while lecture is going on No gossiping while the lecture is going on Raise your hand if you have question to ask Be on time for lecture Be on time to come back from the recess break to continue the lecture Bring your lecturenotes to lecture Kinematics Position Describes motion while ignoring the agents that caused the motion In these lectures we will consider motion in one dimension (along a straight line) We will use the particle model A particle is a point-like object, has mass but infinitesimal size (i.e., the size is very very small) Position is defined in terms of a frame of reference One dimensional, so generally the -ais or y- ais The object s position is its location with respect to the frame of reference The frame can be stationary or moving 3 4

2 Position-Time Graph The position-time graph shows the motion of the particle (car) The smooth curve is an approimation as to what happened between the data points 5 Displacement Defined as the change in position during some time interval Represented as Δ (pronounced as delta ) Δ = f i (final displacement initial displacement) SI units are meters (m), Δ can be positive or negative Displacement may not be always be equal to distance. Distance refers to the length of a path followed by a particle. Give an eample of distance of 5 m but the corresponding displacement is 0. 6 Vectors and Scalars Vector quantities need both magnitude (size or numerical value) and direction to completely describe them We use + and signs to indicate vector directions Scalar quantities are completely described by magnitude only 7 Average Velocity The average velocity is rate (w.r.t. time) at which the displacement occurs Δ f i vaverage = = Δt Δt The dimensions for velocity are length / time, [L/T] The SI units are m/s, or ms -1 It is also the slope of the line (gradient of the tangent line) in the position time graph 8

3 Average Speed Speed is a scalar quantity same units as velocity total distance / total time The average speed is not necessarily the magnitude of the average velocity Give an eample where average speed is m/s and its corresponding velocity is 0. 9 Instantaneous Velocity The limit of the average velocity as the time interval becomes infinitesimally short (very very short), or as the time interval approaches zero The instantaneous velocity indicates what is happening at every point of time. The information includes the direction of movement and magnitude of velocity. 10 Instantaneous Velocity, equations Instantaneous Velocity, graph The general equation for instantaneous velocity is v Δ d = = Δt dt lim0 Δ t (the value of the limit when Δt tends to 0) The instantaneous velocity can be positive, negative, or zero 11 The instantaneous velocity is the slope of the line tangent to the vs. t curve This would be the green line The blue lines show that as Δt gets smaller, they approach the green line Displacement Time 1

4 Some Tangent Values Average Acceleration tan (45 ) = y/ = 1 tan (0 ) = 0 45 y y y Acceleration is the rate of change of the velocity a Δv Δt = = Dimensions are = L/T SI units are m/s² v f L /T T v Δt i tan (90 ) = Instantaneous Acceleration The instantaneous is the limit of the average as Δt approaches 0, which is the value of the constant when Δt tends to 0. a Δ = lim = = Δ t 0 Δt dt dt v dv d Instantaneous Acceleration - graph The slope of the velocity vs. time graph is the The green line represents the instantaneous The blue line is the average 15 16

5 Acceleration and Velocity, 1 When an object s velocity and are in the same direction, the object is speeding up When an object s velocity and are in the opposite direction, the object is slowing down Acceleration and Velocity, The car is moving with constant positive velocity (shown by red arrows maintaining the same size) Acceleration equals zero Acceleration and Velocity, 3 Acceleration and Velocity, 4 Velocity and are in the same direction Acceleration is uniform (blue arrows maintain the same length) Velocity is increasing (red arrows are getting longer) This shows positive and positive velocity 19 Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same length) Velocity is decreasing (red arrows are getting shorter) Positive velocity and negative 0

6 Kinematic Equations - summary Similar set of equations: v = u + at s = s 0 + ½ (u + v)t s = s 0 + ut + ½ at v = u + a (s s 0 ) u : initial velocity s 0 : initial displacement v : final velocity s : final displacement 1 a : We will derive them!! 3 4

7 Kinematic Equations The kinematic equations may be used to solve any problem involving onedimensional motion with a constant Sometime you may need to use two of the equations to solve one problem Many times there is more than one way to solve a problem Kinematic Equations, specific For constant a, We can determine an object s velocity at any time t when we know its initial velocity and its It does not give any information about displacement 5 6 Kinematic Equations, specific For constant, vi + vf v = The average velocity can be epressed as the arithmetic mean of the initial and final velocities Kinematic Equations, specific For constant, This equation gives us the position of the particle in terms of time and velocities The equation doesn t give us the 7 8

8 Kinematic Equations, specific For constant, 1 f = i + vit+ at This equation gives final position in terms of velocity and It doesn t tell you about final velocity Kinematic Equations, specific For constant a, This equation gives final velocity in terms of and displacement It does not give any information about the time 9 30 Graphical Look at Motion displacement time curve (graph) The slope of the curve is the velocity The curved line indicates that the velocity is changing Therefore, there is an What if the curve in the displacement-time graph is a horizontal straight line? displacement time Answer: The object is not moving, i.e., velocity = What if the straight line is not horizontal? Answer: The object is moving with constant velocity, i.e., no. displacement time 3

9 Graphical Look at Motion velocity time curve Graphical Look at Motion time curve The slope gives the The straight line in velocity time graph indicates a constant, which can be 0 if it is an horizontal line. velocity time The zero slope in time graph indicates a constant time Eample: Speedy Sue, driving at 30.0 m/s, enters a one-lane tunnel. She then observes a slow-moving van 155 m ahead traveling at 5.00 m/s. Sue applies her brakes but can accelerate only at.00 m/s because the road is wet. Will there be a collision? If yes, determine how far into the tunnel and at what time the collision occurs. If no, determine the distance of closest approach between Sue's car and the van. - ms - 30 m/s 5 m/s Answer: 30 m/s 5 m/s - ms m 155 m 35 36

10 Freely Falling Objects A freely falling object is any object moving freely under the influence of gravity alone. It does not depend upon the initial motion of the object, which can be: Dropped released from rest Thrown downward Thrown upward Acceleration of Freely Falling Object The of an object in free fall is directed downward, regardless of the initial motion The magnitude of free fall is g = 9.80 m/s g varies slightly at different geographical locations 9.80 m/s is the average at the Earth s surface Acceleration of Free Fall, cont. We will neglect air resistance Free fall motion is constantly accelerated motion in one dimension Let upward be positive, so downward is negative Use the kinematic equations with a y = g = m/s Free Fall Eample Initial velocity at A is upward (+) and is g (-9.8 m/s ) At B, the velocity is 0 and the is g (-9.8 m/s ) At C, the velocity has the same magnitude as at A, but is in the opposite direction The displacement is 50.0 m (it ends up 50.0 m below its starting point) 39 40

11 Motion Equations from Calculus Displacement equals the area under the velocity time curve f lim v Δ t = v ( t) dt Δtn 0 n n n ti The limit of the sum is a definite integral t Kinematic Equations General Calculus Form dv a = dt t v v = a dt f i 0 d v = dt t = v dt f i Kinematic Equations Calculus Form with Constant Acceleration The integration form of v f v i gives vf vi = at The integration form of f i gives 1 f i = vit+ a t v = u + at, or v u = at s = s 0 + ut + ½ at, or s - s 0 = ut + ½ at Answer: Eample: A baseball is hit so that it travels straight upward after being struck by the bat. A fan observes that it takes 3.00 s for the ball to reach its maimum height. Find (a) its initial velocity and (b) the height it reaches. y f y i v f = 0 v i 43 44

12 Eample: Two stones are released from rest at a certain height, one after the other at an interval of c seconds. (a) Will the difference in their speeds increase, decrease, or stay the same? Answer: Let v1 be the speed of first stone, and v the speed of second stone. We have v1 = 0 - gt v = 0 g (t-c); v1 v = -gt + g (t-c) = -gc constant w.r.t. time 45 (b) Will their separation distance increase, decrease or stay the same? Answer: Let s1 be the displacement of first stone, and s the displacement of second stone. We have: s1 = 0 - ½gt for t > 0 s = 0 - ½g (t c) for t > c s1 s = - ½gt + ½g (t c) = - ½gt + ½g (t tc + c ) displacement = ½g ( tc + c ) absolute difference is increasing w.r.t. time Time 46 (c) Will the time interval between the instants at which they hit the ground be smaller than, equal to, or larger than the time interval between the instants of their release? Answer: Let t1 be the time for the first stone to hit the ground, and t the time for the second stone to hit the ground. Let the height be h. We have: -h = 0 - ½gt1 t1 = t1 = h g h g As the distance travelled by the second stone is the same. We have t = c + h g Therefore t t1 = c (same as the difference of the instants of their release.) 47

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