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Constant Acceleration

In terms of our newly-acquired left hemisphere skills, if we use y to designate height [say, above sea level] and t to designate time, then the upward velocity v_y [where the subscript tells us explicitly that this is the upward velocity as opposed to the horizontal velocity which would probably be written v_x]^6.9 is given by

v_y = v_y0 - gt (6.1)

where v_y0 is the initial^6.10 upward velocity (i.e. the upward velocity at t=0), if any,^6.11 and g is the downward^6.12 acceleration of gravity, $g \approx 9.81$ m/s² on average at the Earth's surface.^6.13 Another way of writing the same equation is in terms of the derivative of the velocity with respect to time,

$$a_y \equiv {dv_y \over dt} \equiv \dot{v_y} = - g ,$$ (6.2)

where I have introduced yet another notational convention used by Physicists: a little dot above a symbol means the time derivative of that symbol - i.e. the rate of change (per unit time) of the quantity represented by that symbol.^6.14 And since v_y is itself the time derivative of the height y [i.e. $v_y \equiv dy/dt \equiv \dot{y}$], if we like we can write the original equation as

$$\dot{y} = v_{y_0} - gt.$$ (6.3)

All these notational gymnastics have several purposes, one of which is to make you appreciate the simple clarity of the declaration, ``The vertical speed increases by equal increments in equal times,'' as originally stated by Galileo himself. But I also want you to see how Physicists like to condense their notation until a very compact equation ``says it all.''