Part 4 of Question 2 asks you to compute the acceleration of an object, given th

Question

Part 4 of Question 2 asks you to compute the acceleration of an object, given the change in position, the initial velocity, and the final velocity. To do this, you will need to use both x(t) and v(t) kinematics equations, but ultimately you will eliminate the variable t. Often in mechanics we aren’t particularly concerned about time – we’re only concerned with the change in position, initial velocity, final velocit, and acceleration. These are related by the “third kinematics equation”, v 2 f v 2 0 = 2a(xf x0) Show that this equation is simply a consequence of the other two. Algebra hint: Starting from the x(t) and v(t) formulae for constant acceleration, solve one equation for t and substitute it back into the other one.

Explanation / Answer

x(t) = V0t + 1/2 * at2

and v(t) =V0 + at = > t = [v(t)-V0]/a

so x(t) = V0[v(t)-V0]/a + 1/2 * a{[v(t)-V0]/a}2

=> 2ax(t) = 2V0[v(t)-V0] + {[v(t)-V0]}2 = 2V0v(t)-2V02 + v(t)2+V02-2V0v(t) = v(t)2-V02

thus v(t)2-V02 = 2ax(t)

or   vf2-V02 = 2a[xf -x0] proved

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