Physics problems about work and energy?

Question

I'd greatly appreciate any help, but please show work!
an object with a mass of 0.2 kg is being pushed along a level surface. total displacement is 20 m. the object starts at x=0 m and t=0 sec. the net force varies with the displacement, as shown in a graph. (graph has force in Newtons as y-axis and displacement in meters as x-axis. the line is level at 4 N from 0 to 12 meters. Then it goes from 12 meters at 4 N to 20 meters at 0N)
a) how much work is involved to move it from 0 to 12 meters?
b) what is its speed going from 0 to 12?
c) what is its speed going from 12 to 20?

sorry for not having the graph! thanks for any help

Answer 1

a.) Work = F * d = 4N * 12m = 48 N*m or 48J
b.) Force = m * a
a = F/m = 4/.2 = 20 m/s^2
To find the velocity at any point, use Torricelli's equation:
v^2 = vo^2 + 2ax = 2ax since vo = 0.

At x = 12:
v^2 = 2ax = 2 * 20 * 12 = 480
v = 21.9 m/s

b.) Assuming the force decreases constantly from 12m to 20m, then F = 4 - (x-12)/2
F = ma
a = f/m = (4 - (x-12)/2)/.2 = 20 - 2.5x + 30
a = 50 - 2.5x

v = vo + integral of (a) = 21.9 m/s + (50x - 1.25x^2 | 20, 12)

Note: the 20, 12 is done above, because the acceleration formula is only true from x = 12 to x = 20.

v at 20 meters = 21.9 m/s + (50 * 20 - 1.25 * 400) - (50 * 12 - 1.25 * 144) = 21.9 + (1000 - 500) - (600 - 180) = 101.9 m/s.