4. You do a collision experiment with carts, with expensive equipment that reduces friction with the track to a negligible level. Cart 1 has mass 1.00 kg with initial velocity v₁

= 3.00 m/s, and it heads toward cart 2 with mass 0.500 kg that starts at rest. (a) Obtain an equation that expresses v2f in terms of vif. (b) Obtain an equation for Eloss, the energy lost during the collision, where v₁f is the one remaining unknown. (c) Using an online graphing calculator or similar software (I suggest desmos.com; it's easy to learn), plot both v2f and Eloss as functions of v₁f (which will be the horizontal axis). Focus on the part of the graph where 0 <₁ < 3.5 m/s, and you can see the maximum of Eloss clearly. Save it as a PDF or screen capture; turn in that image. (d) Some of the v1₁ values on your image are impossible to observe in the lab. Why? Give the inequality that defines these impossible velocities. Hint: this is really simple and has nothing to do with Eloss. (e) At what v₁y and v2, values does that maximum Eloss occur? What would you see hap- pening in your experiment? (f) At some possible values of v₁f, you will see that Eloss < 0. How might this happen?