# Potential Energy vs. Position

Consider the following sketch of potential energy for a particle as a function of position. There are no dissipative forces or internal sources of energy. If a particle travels through the entire region of space shown in the diagram, at which point is the particle's velocity a maximum?

# (Math) Plotting Lines for Robots

You're programming the static path for a robot in an enclosed room. The room is represented by a grid with the starting point at the origin (0,0).

After the robot reaches the point (2, -3) represented by the grid, you want it to travel a path perpendicular to another line y = 5x - 13 to avoid crashing into a table.

What's the equation of the line your robot needs to know?

(Original question contributed by Professor Autar Kaw; modified under permission)

Solve!

# (Economics) From Home-Renter to Home-Owner!

You’ve done it. You’ve finally decided to buy a house… sometime in the next 5 years. You’ve got your eye on a certain neighborhood. The homes aren’t any “mansion-in-the-sky.” But it doesn’t matter. Because baby, it’s all yours… (Cough! Well actually it’s the bank’s... at least till you pay it off. But who likes that dream?)

But now that you’ve decided to buy in 5 years, you realize you need to start saving. You figure if home prices continue their uptrend, you’re dream house will probably cost about 325,000 USD once you’re ready. The average cash down-payment typically runs about 20% of the purchase price. And for the foreseeable future, it seems reasonable that you can get 7% annual return (compounded monthly) if you invest your savings in a mutual fund.

If you choose to automatically deposit money every month into such a fund, how much do you need to save each month in order to afford the down-payment for this house?

FOUNDATIONAL TOOL / HINT: So this is a jump off a cash-flow concept we’ve explored in the past… but this time, from a different angle. If you haven’t explored this or need a refresher, CLICK HERE to revisit the explanation. But if you just need a simple Hint, then CLICK HERE

# (Thermo) Energy In & Out Of A Stirling Engine

You've just gotten your hands on a stirling engine (Dean Kamen thinks these things could power the world). The engine takes an input from a heat source and outputs work.

In this case, a pressure sensor tells us that the heat source compresses the (ideal) gas inside the piston chamber from 200 kPa to 300 kPa. As a result of the process, the internal energy of the gas increases by 10 kJ, and 140 kJ of heat is transferred to the surroundings.

How much work was done by the gas during this part of the process?