Please show work, equations and units! Thank you!

  0.56 m/s

Explanation:

The speed of the head at the end of the interval in each case is the area under the acceleration curve. Then the difference in speeds is the difference in areas.

We can find the area geometrically, using formulas for the area of a triangle and of a trapezoid.

  A = 1/2bh . . . . area of a triangle

  A = 1/2(b1 +b2)h . . . . area of a trapezoid

If h(t) is the acceleration at time t for a helmeted head, the area under that curve will be (in units of mm/s) ...

  Vh = 1/2(h(3)·3) +1/2(h(3) +h(4))·1 +1/2(h(4) +h(6))·2 +1/2(h(6))·1

  Vh = 1/2(4h(3) +3h(4) +3h(6)) = 1/2(4·40 +3·40 +3·80) = 260 . . . mm/s

If b(t) is the acceleration for a bare head, the area under that curve in the same units is ...

  Vb = 1/2(b(2)·2 +1/2(b(2) +b(4))·2 +1/2(b(4) +b(6))·2 +1/2(b(6)·1)

  Vb = 1/2(4b(2) +4b(4) +3b(6)) = 1/2(4·120 +4·140 +3·200) = 820 . . . mm/s

Then the difference in speed between the bare head and the helmeted head is ... (0.820 -0.260) m/s = 0.560 m/s