Position-time graphs of cars A and B

Car A and Car B have position-time graphs that are straight lines making angles of 30 degrees and 60 degrees with the time axis, respectively. In order to find the ratio of velocities of A and B, we need to analyze the slopes of their position-time graphs.

Slope of the position-time graph

The slope of a position-time graph represents the velocity of an object. It is the rate at which the position of the object changes with respect to time. Mathematically, the slope can be calculated as the change in position divided by the change in time.

Calculating the slope of the position-time graph for Car A

The angle made by the position-time graph of Car A with the time axis is 30 degrees. Let's assume the slope of Car A's graph is m1.

tan(30 degrees) = m1
1/sqrt(3) = m1

Therefore, the slope of Car A's position-time graph is 1/sqrt(3).

Calculating the slope of the position-time graph for Car B

The angle made by the position-time graph of Car B with the time axis is 60 degrees. Let's assume the slope of Car B's graph is m2.

tan(60 degrees) = m2
sqrt(3) = m2

Therefore, the slope of Car B's position-time graph is sqrt(3).

Ratio of velocities of A and B

The ratio of velocities of Car A and Car B can be calculated by dividing their respective slopes.

Velocity of Car A / Velocity of Car B = m1 / m2
Velocity of Car A / Velocity of Car B = (1/sqrt(3)) / sqrt(3)
Velocity of Car A / Velocity of Car B = 1/3

Therefore, the ratio of velocities of Car A and Car B is 1:3.

Explanation

The ratio of velocities of Car A and Car B is determined by the slopes of their position-time graphs. The slopes represent the rates at which their positions change with respect to time. By calculating the slopes using the angles made by their graphs with the time axis, we can determine the ratio of their velocities. In this case, the ratio is 1:3, indicating that Car B is three times faster than Car A.