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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."