Straight-Line Motion:Connecting Position, Velocity, and Acceleration

In straight-line (one-dimensional) motion, the position of a moving object changes over time along a linear path.

The behavior of such motion can be modeled using three core functions in calculus:

Position function:

\( s(t) \) : describes the exact location (displacement) of an object along a straight line at time \( t \), typically measured in meters (m).

Velocity function:

\( v(t) = \dfrac{ds}{dt} \) : represents the rate of change of the position with respect to time. It tells both the speed and the direction of motion. Positive velocity indicates forward motion; negative velocity indicates backward motion.

Acceleration function:

\( a(t) = \dfrac{dv}{dt} = \dfrac{d^2s}{dt^2} \) : represents the rate of change of velocity with respect to time. It shows how the speed and/or direction of motion is changing.

Relationship Summary:

\( v(t) = \dfrac{ds}{dt} \quad \text{and} \quad a(t) = \dfrac{dv}{dt} = \dfrac{d^2s}{dt^2} \)

Concepts and Interpretation

• When \( v(t) > 0 \): The object is moving in the positive direction (right/forward).
• When \( v(t) < 0 \): The object is moving in the negative direction (left/backward).
• When \( v(t) = 0 \): The object is momentarily at rest.
• When \( a(t) > 0 \): The velocity is increasing; the object is speeding up (if already moving forward) or slowing down (if moving backward).
• When \( a(t) < 0 \): The velocity is decreasing; the object is slowing down (if moving forward) or speeding up (if moving backward).

Speeding Up vs. Slowing Down:

• The object is speeding up when velocity and acceleration have the same sign.
• The object is slowing down when velocity and acceleration have opposite signs.

Units and Real-World Meaning

• Position, \( s(t) \): measured in distance units like meters (m).
• Velocity, \( v(t) \): measured in meters per second (m/s); unit for position divided by time.
• Acceleration, \( a(t) \): measured in meters per second squared (m/s²); unit for velocity divided by time.

Example Interpretation: If \( v(3) = -5 \) and \( a(3) = -2 \), then the object is moving left at 5 m/s and speeding up (since velocity and acceleration are both negative).

Graphical Interpretation

• The graph of \( s(t) \) (position vs. time) gives insight into the motion of the object.
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• The slope of the tangent to the graph of \( s(t) \) at a point gives the velocity \( v(t) \).
• The slope of the graph of \( v(t) \) gives the acceleration \( a(t) \).
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• Points where \( v(t) = 0 \) correspond to turning points or rest points in the motion.