Final answer:
In a weighted graph, edge lengths are not necessarily proportional to their weights, making the statement False. The Pythagorean theorem can be used to find the length of the resultant vector of two perpendicular vectors (True), and the position vs time graph of an accelerating object is not a straight line (False).
Step-by-step explanation:
The statement that in a weighted graph, the lengths of the edges are proportional to their weights is False. The weight of an edge in a graph tends to represent the cost or capacity of that edge, not its physical length. In graph theory, a weighted graph is simply one where the edges have weights or values associated with them. These weights can represent various things depending on the context, such as the cost of traveling between two nodes, the distance, or the time it might take to travel from one node to another. However, there is no requirement that the visual representation of the graph reflects these weights in terms of the lengths of the edges drawn on a page or screen.
Now, let's address the Pythagorean theorem. The statement is True that we can use the Pythagorean theorem to calculate the length of the resultant vector obtained from the addition of two vectors which are at right angles to each other. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. When two vectors are at right angles, they form the two sides of a right-angled triangle, with the resultant vector being the hypotenuse.
Regarding the position vs time graph of an object that is speeding up, the statement is False. An object that is speeding up is one that is accelerating, and the graph of its position over time would not be a straight line but rather a curve that gets steeper as time progresses. Indeed, a straight line on a position versus time graph signifies a constant velocity, not an increasing velocity.
The amplitude of waves is another interesting topic. It is False that the amplitudes of waves add up only if they are propagating in the same line. Waves can interfere with each other when they meet; this is known as superposition. When two waves meet, their amplitudes can add (constructive interference) or subtract (destructive interference) at the point of intersection, regardless of the direction in which they are propagating.
Similarly, the statement that the amplitude of one wave is affected by the amplitude of another wave only when they are precisely aligned is False. Again, due to the principle of superposition, waves can affect each other's amplitudes when they intersect or overlap, independent of their alignment.
For the scenario where a rock is thrown into the air, the statement is False. As a rock is thrown up and gains height, it loses kinetic energy and gains potential energy due to gravity. It's only as the rock falls back towards the ground that it gains kinetic energy and loses potential energy.
Lastly, considering the circuit diagram, it is False that we can assume the voltage is the same at every point in a given wire. In a circuit, the voltage can drop across components like resistors, and it is not the same at every point unless it is an ideal wire with no resistance and no voltage drop across it.