Displacement, Velocity, and Acceleration

Learning Goals

After this lesson, you should be able to:

• use the object model to represent a moving object as a point;
• calculate displacement from initial and final positions;
• distinguish average velocity from average speed;
• calculate average acceleration from a velocity change;
• interpret positive and negative velocity and acceleration consistently; and
• explain how a short-interval average approaches an instantaneous value.

1. The Object Model and a Coordinate Choice

When an object's size, shape, and rotation do not affect the question, model the object as a single point. Choose an origin and one positive direction before assigning signs. The model does not claim the real object has no size; it deliberately omits details that are unnecessary for tracking translational motion.

A complete setup states the object, reference frame, origin, positive direction, initial time, and initial position.

2. Position and Displacement

Position \(x\) is the object's signed coordinate relative to the origin. Displacement is a change in position, not the coordinate itself and not the total path length.

The SI unit is the meter. If \(x_i=+6\,\mathrm m\) and \(x_f=-2\,\mathrm m\), then \(\Delta x=-8\,\mathrm m\). The negative sign means the net change points in the negative axis direction.

3. Average Velocity and Average Speed

Average velocity uses net displacement. Average speed uses total distance traveled. They answer different questions when an object reverses direction.

Elapsed time \(\Delta t\) is positive for a forward-running clock. Average velocity may be positive, negative, or zero; average speed is nonnegative. Both have SI unit \(\mathrm{m\,s^{-1}}\).

4. Average and Instantaneous Acceleration

Average acceleration compares two velocity states over a time interval.

Its SI unit, \(\mathrm{m\,s^{-2}}\), means that velocity changes by a certain number of meters per second during each second. Instantaneous velocity or acceleration describes one instant. Experimentally, it can be estimated by calculating the corresponding average over successively shorter intervals surrounding that instant.

6. Velocity and Acceleration Signs

The reliable rule is: speed increases when velocity and acceleration point in the same direction and decreases when they point in opposite directions. Negative acceleration does not automatically mean slowing down.

7. Worked Example: Displacement and Average Velocity

Question: A cart is at \(x_i=-4.0\,\mathrm m\) when \(t_i=1.0\,\mathrm s\). At \(t_f=4.0\,\mathrm s\), it is at \(x_f=+8.0\,\mathrm m\). Find its displacement and average velocity.

Answer: The displacement is \(+12.0\,\mathrm m\), and the average velocity is \(+4.0\,\mathrm{m\,s^{-1}}\). These endpoint data do not prove that the cart moved at a constant velocity between the two times.

8. Worked Example: Acceleration Through a Reversal

Question: A cart's velocity changes from \(+6.0\,\mathrm{m\,s^{-1}}\) to \(-2.0\,\mathrm{m\,s^{-1}}\) in \(4.0\,\mathrm s\). Find its average acceleration.

Answer: The average acceleration is \(-2.0\,\mathrm{m\,s^{-2}}\). The cart initially slows while moving in \(+x\), reaches zero velocity if the change is continuous, and then speeds up in \(-x\). A negative acceleration alone does not describe all three stages.

9. AP Comparison and Justification

Two runners have the same average velocity over \(10\,\mathrm s\). Runner A moves steadily; Runner B changes speed several times but reaches the same final position from the same initial position.

Claim: Their instantaneous velocities need not be equal at any selected intermediate time. Evidence: average velocity depends only on total displacement and elapsed time. Reasoning: many different time-by-time velocity histories can produce the same net displacement over the same interval.

10. Investigation: From Video to Motion Data

Place evenly spaced markers beside a straight track and record a cart from a stationary side view. Choose the track direction as the \(x\)-axis. From video frames, record position and time in a table.

1. Calculate \(\Delta x/\Delta t\) for several adjacent frame intervals.
2. Repeat using wider intervals and compare the averages.
3. Estimate the velocity near one chosen instant using the narrowest reliable interval.
4. Calculate changes between successive velocity estimates to infer the acceleration direction.
5. State how marker spacing, frame rate, camera angle, and position uncertainty affect the conclusion.

Use original measurements rather than assuming the cart has constant acceleration.

11. Common Traps and Fast Checks

• Trap: using \(x_f+x_i\) for displacement. Check: change always means final minus initial.
• Trap: dividing distance by time and calling it average velocity after a reversal. Check: velocity requires displacement.
• Trap: interpreting \(a_x<0\) as “slowing down.” Check: compare the signs of \(v_x\) and \(a_x\).
• Trap: treating an interval average as an exact instantaneous value. Check: identify whether the question names an interval or an instant.
• Trap: reporting acceleration in \(\mathrm{m\,s^{-1}}\). Check: acceleration units are velocity change per time, \(\mathrm{m\,s^{-2}}\).

12. Key Takeaways

• Displacement compares positions: \(x_f-x_i\).
• Average velocity is displacement per elapsed time.
• Average acceleration is velocity change per elapsed time.
• Instantaneous quantities are approximated by averages over very short intervals.
• Velocity and acceleration signs describe directions; their relationship determines whether speed increases or decreases.