Question

1. consider the position vs. time graph below for cyclists a and b.

a. do the cyclists start at the same point? how do you know? if not, which is ahead?
b. at t = 7s, which cyclist is ahead? how do you know?
c. which cyclist is travelling faster at t = 3s? how do you know?
d. are their velocities equal at any time? how do you know?
e. what is happening at the intersection of lines a and b?

Explanation:

Step1: Check starting - point

On a position - time graph, the starting position is at \(t = 0\). Looking at the graph, cyclist A starts at a non - zero position while cyclist B starts at the origin (\(x=0\)). So they do not start at the same point, and cyclist A is ahead at the start.

Step2: Check position at \(t = 7s\)

Draw a vertical line at \(t = 7s\) on the graph. Observe the intersection of this line with the graphs of A and B. The position of cyclist A is higher (greater \(x\) - value) than that of cyclist B at \(t = 7s\), so cyclist A is ahead.

Step3: Check speed at \(t=3s\)

Speed is given by the slope of the position - time graph. At \(t = 3s\), the slope of the graph of cyclist A is steeper than that of cyclist B. A steeper slope means a higher speed, so cyclist A is travelling faster at \(t = 3s\).

Step4: Check equal velocities

Velocities are equal when the slopes of the two graphs are equal. The slopes of the graphs of A and B are not equal at any time (the lines are straight and have different slopes), so their velocities are not equal at any time.

Step5: Analyze intersection

At the intersection of lines A and B, the two cyclists are at the same position at that instant of time.

Answer:

a. No, they do not start at the same point. Cyclist A starts ahead as it has a non - zero initial position while cyclist B starts at \(x = 0\).
b. Cyclist A is ahead at \(t = 7s\) as its \(x\) - value is greater at the intersection of \(t = 7s\) with its position - time graph.
c. Cyclist A is travelling faster at \(t = 3s\) since the slope of its position - time graph is steeper at that time.
d. No, their velocities are not equal at any time because the slopes of their position - time graphs are not equal at any point.
e. At the intersection of lines A and B, the two cyclists are at the same position at that time.