A data pair (x, y) follows y = m x + b. If x represents time and y represents position, what does the slope m represent in a physical context of motion?

• A. The intercept b
• B. The acceleration
• C. The velocity at x = 0
• D. The rate of change; e.g., velocity if y is position and x is time

Answer: (D)

When time is the independent variable and position is the dependent variable, the slope m tells how quickly position changes as time passes. That rate of change is velocity. In a linear relation y = m x + b, if x represents time, then the slope m is the velocity, since it equals dy/dt. The larger the slope, the faster position changes per second; the units work out to meters per second (assuming y is in meters and x in seconds). The intercept b simply represents where the object is at time zero, not how fast it’s moving. Acceleration, on the other hand, is the rate at which velocity itself changes with time, which would require the slope to change with x; here the slope is constant, so acceleration is zero in this straight-line motion.