Tutor profile: Eric A.
Subject: Java Programming
Name three differences between an interface and an abstract class.
There are many possible answers here, but some of the most prominent are: A class can implement multiple interfaces, but only extend one abstract class. An abstract class can implement a constructor, while an interface has no such thing. An abstract class may have private fields, while an interface has no such thing.
Subject: Discrete Math
The four color theorem is a well known result in graph theory stating that any planar graph has a chromatic number of at most four, that is, that you need at most four different colors to assign a color to each vertex so that no two adjacent vertices have the same color. Show that the converse isn't true, that a graph having a chromatic number of 4 or less doesn't imply that the graph is planar.
To disprove the claim "If a graph has a chromatic number of 4 or less, it must be planar", all we need to do is show a counterexample. We therefore need to think of a graph which isn't planar, but does have a chromatic number of 4 or less. The simplest graphs we know to be nonplanar are the complete graph K5 and the complete bipartite graph K3,3. Of these, K5 isn't any help, as it has a chromatic number of 5. However, by definition any bipartite graph has a chromatic number of 2, so K3,3, despite having a chromatic number of 2, is non-planar and disproves the claim above.
Clara, 60 kg, is skating west on a frictionless surface at 2 m/s. Her 10 kg dog runs up to her in the opposite direction at 10 m/s and jumps into her arms. What speed and direction do the two end up going in?
As with many mechanics problems, the key to this one is identifying a conserved quantity, in this case linear momentum. Recall that the formula for linear momentum is just mass times velocity, so Clara has momentum of 60 kg * 2 m/s = 120 kg m/s. Her dog has momentum 10 kg * -10 m/s = - 100 kg m/s, where the minus sign comes from the dog's velocity being in the opposite direction. The dog jumping into her arms is a perfectly inelastic collision, so the two continue on together with total momentum conserved of 120 - 100 = 20 kg m/s. Together the two weigh 60 + 10 = 70 kg, so in the end they will be travelling at 20 / 70 = 2/7 m/s, or approximately 0.28 m/s.
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