Kinematics

Consider a particle moving along a straight line BOA (Fig. 3.1). The position of the particle is given by its distance from O, distances to the right of O being taken to be positive and distances to the left of O being taken to be negative. The particle is at the point P, where OP = s, at time t and at the point Q, where OQ = s + δs at time t + δt. Thus, in moving from P to Q, the particle travels a distance δs in time δt. The average velocity of the particle over this interval of time δt is <math display='block'> <mrow> <munder> <mrow> <mi>lim</mi> </mrow> <mrow> <mi>&#x03B4;</mi><mi>t</mi><mo>&#x2192;</mo><mn>0</mn> </mrow> </munder> <mfrac> <mrow> <mi>&#x03B4;</mi><mi>s</mi> </mrow> <mrow> <mi>&#x03B4;</mi><mi>t</mi> </mrow> </mfrac> <mo>,</mo><mtext>&#x2009;</mtext><mtext>&#x200A;</mtext><mi>i</mi><mo>.</mo><mi>e</mi><mo>.</mo><mtext>&#x2009;</mtext><mfrac> <mrow> <mi>d</mi><mi>s</mi> </mrow> <mrow> <mi>d</mi><mi>t</mi> </mrow> </mfrac> </mrow> </math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\[\mathop {\lim }\limits_{\delta t \to 0} \frac{{\delta s}}{{\delta t}},\;{\kern 1pt} i.e.\;\frac{{ds}}{{dt}}\]$$.