Linear Motion

 Kinematics - Linear Motion

1.1  Linear motion and Non-Linear Motion:

·        Linear motion is motion in a straight line – For examples: A passenger is carried by an escalator; or an athlete running a 100 m race; whereas

·        Non-linear motion is motion not in a straight line - A top spinning; or the earth orbiting the Sun.

1.2  Kinematics and Dynamics:

·        Kinematics is the study of the motion of an object without considering the force acting on it. Therefore, the equations of motion do not have force F as a variable in them.

·        Dynamics is the study of motion and the forces acting on the object.

1.3  The physical quantities involved in linear motion:

·        Distance and Displacement

·        Speed and Velocity

·        Acceleration and Deceleration

·        Time

The Physical Quantities Explained:

Suppose an object at point O moves east 100 metres (m) to point A in 14 seconds (s); then it immediately moves west from point A to point B 40 m away in 8 s:

·        Distance:

o       The total distance moved by the object = (100 + 40) m = 140 m;

o       Distance is the total length of the path travelled by an object in motion from 1 position to another position;

o       Distance is a scalar quantity – it does not take into account the direction of motion.

o       In linear motion: Distance = Magnitude of Displacement (since direction of motion remains constant)

·        Displacement:

o       The total displacement of the object from point O = (100 – 40) m = 60 m east of point O;

o       Displacement is a measure of how far and the direction in which an object has been displaced from a reference point (e.g. original position) due to the motion

o       Displacement measures the straight-line distance and the direction between the initial position and the final position of an object due to the motion.

o       Displacement is a vector quantity because it has both the magnitude and the direction.

·        Speed:

o       The object’s Average Speed from O to B = 140 m / 22 s = 6.36 m s^-1:

§         Its Av. Speed from O to A = 100 m / 14 s = 7.14 m/s

§         Its Av. Speed from A to B = 40 m / 8 s = 5 m/s

o       Thus, Average Speed, v, = Total Distance Travelled, s (m) / Time Taken, t (s).

o       Constant Speed: If an object moves equal distances in equal time intervals, then it is moving with constant speed – otherwise, it is moving with non-uniform speed.

o       Speed is a measure of how fast an object moves – the rate of distance travelled in the motion.

o       Speed is a scalar quantity – it measures only the magnitude of distance moved over time with no regard to direction of motion.

o       In linear motion: Speed = Magnitude of Velocity (since the direction of motion remains constant)

·        Velocity:

o       Its average velocity from O to B = (100 – 40) m / 22 s or 2.73 m/s  due east of O:

§         Its Av. Velocity from O to A = 100 m / 14 s = 7.14 m/s due east of O;

§         Its Av. Velocity from A to B = - 40 m / 8 s = - 5 m/s (west of A)

o       Thus, Average Velocity, v = Total Displacement, s (m) / Time Taken, t (s).

o       Constant Velocity v means equal displacements in equal time intervals.

o       Constant Velocity: If an object moves with equal displacement in equal time intervals, then it is moving with constant velocity – otherwise, it is moving with non-uniform velocity.

o       Velocity is a measure of how fast an object is displaced (the rate of displacement of the object) from a reference position (its initial position).

o       Velocity, unlike speed, is a vector quantity because it measures both the magnitude and direction of displacement due to motion.

o       A change in velocity means either a change in speed or a change in direction (or both) of motion.

·        Speed and Velocity in Linear Motion:

o       In a linear motion, the direction of motion remains unchanged: The speed of linear motion equals to the magnitude of velocity since the distance travelled per unit time in a linear motion is the same as the displacement per unit time along the straight line. Thus, in linear motion:

v = v = Total distance travelled, s (m) / Time taken, t (s)

o       In linear motion, a change in velocity can only means a change in the speed with no change in direction of motion and this change of speed can only be due to acceleration (increase in speed over time) or deceleration (decrease in speed over time).

·        Acceleration and Deceleration:

o       Acceleration is defined as the rate of change of velocity (or rate of change of speed, in linear motion):

Acceleration, a = Change in Velocity / Time Taken

                         = (Final Velocity, v – Initial Velocity, u) / Time, t

                      a = (v – u) / t …1st Equation of Motion

o       Acceleration is positive if initial velocity, u, increases with time, t, to final velocity, v: that is, if v > u, a is +ve. 

o       Conversely, acceleration is negative if velocity decreases to final velocity, v, from initial velocity, u: that is, if v < u, a is –ve, in which case, it is known as deceleration or retardation.

o       Constant (or uniform) acceleration:

·        If an object moves with equal change in velocity (or speed in linear motion) in equal time intervals, then it is moving with constant acceleration – otherwise, it is moving with non-uniform acceleration.

·        An object moves with constant acceleration, a, for time, t, will change the initial velocity, u, to final velocity, v, as follows:

v = u + at …2nd Equation of Motion; Therefore,

t = (v – u) / a …3rd. Equation of Motion 

o       Zero acceleration: If velocity is constant or uniform (or, speed is constant or uniform in the case of linear motion), the acceleration is zero since change in velocity (or, speed) is zero.

o       Acceleration or deceleration is a vector quantity with SI unit metre per second per second, m s^-2.

o       Equations of Motion are derived on the basis of constant or zero acceleration – please see later.

·        Time (Scalar Quantity in second (s)):

·        Time interval is an important quantity in the study of motion – it is measured by a stopwatch or by the use of “Ticker-Timer”.

·        There are 2 types of stopwatches depending of the accuracy needed:

Accuracy Needed              Type of Stopwatch Used

0.1 s ~ 0.2 s                       Analogue (mechanically-operated)

0.01 s                                 Digital (electronically-operated)

·        Apart from stopwatch, another device that we use to measure time interval of linear motion is the “ticker-timer” – please see the ensuing.

·        We know that an object that moves with constant acceleration, a, for time, t, will change its initial velocity, u, to final velocity, v:

v = u + at … 2nd Equation of Motion (as in foregoing);

         Therefore, time, t = (v – u) / a …3rd Equation of Motion

·        Ticker-Timer

Ticker-timer is a device that can be used to determine a number of quantities relating to linear motion of an object (including time interval), namely:

a)      Time Interval of the Motion

b)      Displacement of the Object

c)      Velocity of the Object

d)      Acceleration of the Object

e)      Type of Motion of the Object

·        The above quantities can be determined because ticker-timer has a metal strip with a pin that vibrates up and down at 50 Hz (which is the frequency of the 12 V or 6 V ac power supply). Each time the pin moves down at the interval of 1/50 seconds (or 0.02 s), it makes a dot on the pre-carbonated ticker tape which passes beneath it as the tape is pulled by the moving object to which the tape is attached. Thus,

a)      The time interval which elapses between successive dots is 1/50 second or 0.02 s (T = 1/f) – therefore, the time interval of the motion between any 2 points / dots can be determined by: Multiplying the number of dots after the 1^st point / dot until the other point / dot by 1/50 s; Therefore:

1 dot-space (1-tick space) = 1 x 0.02 s = 0.02 s

2 dot-space (2-tick of time) = 2 x 0.02 s = 0.04 s

5 dot-space (5-tick of time) = 5 x 0.02 s = 0.1 s

10 dot-space (10-tick space) = 10 x 0.02 s = 0.2 s

b)      Displacement of the object between any 2 points can also be determined by measuring the distance between the 2 points on the ticker tape

c)      Velocity:

·        The velocity of the objects between 2 points / dots is the displacement between the 2 points / dots over the number of tick-time over the 2 points / dots;

·        The average velocity between any 2 points is the total displacement between the 2 points of the tape over the total time intervals elapsed (i.e. the number of tick-time) between the 2 points.

·        Constant (or, Uniform) Velocity: When the ticker tape shows equal displacement over equal time intervals, the object is moving linearly with constant or uniform velocity.

d)      Acceleration or Deceleration:

·        The acceleration (or deceleration) of the objects between 2 successive intervals of motion on the ticker-tape is the change in velocities between the 2 intervals over the time between the mid-points of the 2 intervals. The velocities can be determined as in the foregoing.

·        The average acceleration (or deceleration) of a moving object between any 2 intervals of motion (note: intervals not points) can be determined by comparing the corresponding velocities for the 2 intervals over the time between the mid-points of the 2 intervals - increasing velocity means acceleration and decreasing velocity means deceleration.

·        Constant (or uniform) Acceleration: When the ticker tape shows equal increase (or decrease) in velocity over equal time interval, the object is moving linearly with constant acceleration (or constant deceleration, which ever is appliacable)

e)      From the foregoing, it is therefore clear: That the type of motion of an object – whether it is moving linearly at constant velocity (zero acceleration), at irregular acceleration or deceleration or at constant acceleration– can be seen and determined from the pattern of the dots on the ticker tape.

·        Equations of Motion

We have already learnt from the foregoing 3 basic equations of linear motion where the object moves with constant (or uniform) acceleration, a, with initial velocity, u, final velocity, v, for time, t:

1)      a = (v – u) / t …1^st  Equation of Motion

2)      v = u + at …2^nd Equation of Motion; and

3)      t = (v – u) / a …3^rd. Equation of Motion 3

The 4^th Equation of Motion is about displacement or distance travelled, s, of the object under the same state of motion:

4)      s = Average Velocity x Time Taken

      s = ½ (u + v)t … 4^th Equation of Motion

     

The 5^th and 6^th Equations of Motion are obtained from the 4^th Equation [s = ½ (u + v)t] by substituting the 2^nd Equation (v = u + at) and the 3^rd equation [t = (v - u)/a] respectively into the 4^th Equation. Thus,

5)   s = ½ (u + v)t…from 4^th Equation

s = ½ [u + (u + at)]t …(substitute 2^nd equation into the 4^th)

s = ½ [2u + at]t

s = ut + ½ at^2 …5^th Equation of Motion

(or, s = 1/2 (g) (t^2), free fall under gravity: 2005 P1 Q4. pg 3)

6)      s = ½ (u + v)t…from 4^th Equation

      s = ½ (u + v)(v – u)/a …(substitute 3^nd equation into the 4^th)

      s = ½ (v^2 – u^2)/a …(Form 3 algebraic expansion)

2as = v^2 – u^2

 v^2 = u^2 + 2as … 6^th Equation of Motion

Science is best learnt by understanding rather than by memorizing formulae (rote learnig). I believe, by understanding alone, you should be able to easily derive the first 4 equations of motion. In solving a kinematics problem, just ask yourself this: What is the variable that the question wants me to find the value?

·        If acceleration, a: then, use 1^st equation;

·        If final velocity, v: use 2^nd equation;

·        If time, t: use 3^rd equation;

·        If displacement, s: use 4^th equation.

At times, you need to find the value of another variable before you can solve your problem using the above simplified method – ample examples in my handouts.