Question

6a) if the velocity is negative, the man is walking towards the tree under all conditions
true
false
6b. if the velocity is positive, the mans position is always greater than zero.
true
false
6c. a negative velocity means the man is slowing down (speed is getting smaller).
true
false

Explanation:

• 6a: Velocity is a vector quantity. The sign of velocity indicates direction. If we assume the direction towards the tree is a certain reference (say positive), then negative velocity would be the opposite direction. But if the reference is set such that negative is towards the tree (for example, if we define the coordinate system with the tree at a negative - position side), then negative velocity is towards the tree. However, generally, we assume a standard coordinate system (e.g., right - hand coordinate system in 1D where positive is a default direction). But the problem says "under all conditions". Since the direction of what negative velocity represents depends on the coordinate system definition, it's not always towards the tree.
• 6b: Velocity being positive means the direction of motion is in the positive - defined direction. But the initial position could be negative. For example, if a man starts at \(x=- 5\) and has a positive velocity \(v = 2\), his position function \(x(t)=x_0+vt=-5 + 2t\). When \(t = 1\), \(x(1)=-3\) (still negative). So, just because velocity is positive doesn't mean position is always greater than zero.
• 6c: Velocity is \(v\) and speed is \(|v|\). A negative velocity just means the direction of motion is opposite to the positive - defined direction. For example, if \(v=-3\) (constant), the speed \(|v| = 3\) (constant, not slowing down). Acceleration (the rate of change of velocity) determines if an object is slowing down (if \(a\) and \(v\) have opposite signs) or speeding up (if \(a\) and \(v\) have the same signs).

Answer:

6a. B. False
6b. B. False
6c. B. False