Two boxes of equal mass are released from the same point on a ramp. The green box (Box A) slides without friction down the ramp. The red box (Box B) has a kinetic force of friction acting on it.
Predict what the graphs of gravitational potential energy, kinetic energy, and total mechanical energy look like for each box as a function of distance from the starting point. The zero level for potential energy is the bottom of the ramp.
Predict what the graphs look like as a function of time.
The first thing to consider is what shape the graphs should have. Let's say the boxes start off at a height h above the bottom of the ramp. If we graph energy with respect to distance from the starting point we have:
U(x) = mgh - mgx*sin(θ)
Work done by friction = -fkx
K(x) = mgx*sin(θ) - fkx
Total mechanical energy is E(x) = mgh - fkx
These are the general equations. The equations for box A are simpler because there is no friction.
These are all linear.
Graphed versus time the graphs are all quadratic. The force is constant for each box, and each box starts from rest, so:
x = 0.5 at2
Substitute this for x in the equations above to find expressions for U(t), K(t), and E(t), which will all be quadratic in t except for the total mechanical energy for box A, which is constant.