Core Unit I: Kinematics and Dynamics
E. Acceleration

Key Concepts

Acceleration is the rate of change of an object's velocity.

Acceleration is a vector quantity.

The SI unit for acceleration is (m/s)/s or m/s2.

An object undergoes constant, or uniform, acceleration if its velocity changes by equal amounts in equal time intervals.

Average acceleration is the change in an object's velocity during some elapsed time .

The acceleration of an object at a specific time is called the instantaneous acceleration.

A velocity versus time graph can be used to analyze an object's motion. The slope gives the object's acceleration. The area under the curve gives the displacement. The slope of a line segment joining two points on the graph gives the average acceleration during that interval. The slope of the tangent to the curve at a particular point in time gives the instantaneous acceleration at that time.

Slope can be found on a velocity versus time graph by dividing the change in velocity by the change in time.

Negative acceleration is sometimes referred to as deceleration.

An object travelling at a constant velocity has an acceleration of zero.

Some useful equations for uniform acceleration are:

A velocity versus time graph can be used to show that:

The graph should also be used to derive the equations for uniform acceleration. (Use graphical analysis whenever possible.)

When solving problems involving uniform acceleration, a careful analysis of the given information can help to determine which equations could be used.

An object rolling down an inclined plane has a constant acceleration. The slope of the incline determines the magnitude of the acceleration.

Learning Outcomes

Students will increase their abilities to:

1. Define the following terms: acceleration, average acceleration, instantaneous acceleration.

2. State the SI units for displacement, velocity, and acceleration.

3. Distinguish between uniform and non-uniform acceleration.

4. Give examples of objects undergoing constant acceleration.

5. Determine the average velocity of an object graphically and algebraically.

6. Estimate the instantaneous acceleration of an object graphically.

7. Distinguish between positive and negative acceleration.

8. Recognize situations which illustrate an acceleration of zero.

9. Analyze velocity versus time graphs to determine acceleration, average acceleration, and instantaneous acceleration.

10. Analyze velocity versus time graphs to determine an object's displacement during specified time intervals.

11. Interpret velocity versus time graphs to determine the velocity of an object at specific instants in time.

12. Obtain instantaneous accelerations from a velocity versus time graph and use them to develop an acceleration versus time graph.

13. Recognize that the equations for uniformly accelerated motion can be derived from first principles.

14. Solve problems involving acceleration using the equations for uniformly accelerated motion.

15. Use a velocity versus time graph to develop a displacement versus time graph and an acceleration versus time graph.

16. Use graphs to analyze various kinds of physical phenomena.

17. Interpret and apply ratios, proportions, percentages, and other mathematical concepts correctly.

18. Relate an understanding of acceleration to familiar experiences and practical applications.

Teaching Suggestions, Activities and Demonstrations

1. Connect a recording timer tape to the bob of a pendulum. Allow the pendulum to swing through half of a complete cycle, from one extreme to the other. Develop a velocity versus time graph from the data collected on the recording tape. Determine inflection points on the graph. Find the position(s) of the bob which produce maximum acceleration.

2. Construct an accelerometer for use in analyzing accelerated motion. Several possible designs are shown below:
   • Tank filled with a thin water wedge between two glass plates.

     

     A liquid-surface accelerometer can be used to measure acceleration directly. If the length of the accelerometer is 19.6 cm, the height of the liquid above or below the rest position will give the acceleration in m/s2.

   • Inertial accelerometer. The displacement of the needle on the calibrated scale is in the opposite direction to the acceleration.

     

   • Mason-jar accelerometer.

     

   • Combination accelerometer and pendulum mounted on opposite sides of a rotating serving tray. As the apparatus rotates, the pendulum bob swings outward and the jam jar bob swings inward. Students can try to account for the anomalous behaviour of the two bobs.

     

3. Perform an activity to analyze the motion of an object undergoing uniform acceleration.

4. To perform ramp experiments, use grooved wooden moulding half-rounds, available from building suppliers, and steel balls. "Races" can be held with the moulding at different angles. If the moulding is long enough, curve it. Compare the motion of a ball following the curved path and the straight- line path. Even though the shortest distance between two points may be a straight line, the shortest time needed to travel between those two points occurs when one follows the longer, curved path. This is an intriguing anomaly.

5. Roll an empty cylindrical metal coffee can down an inclined plane. Have the students predict what would happen if a similar can, completely filled with sand, were rolled down at the same time as the empty one. Students who are familiar with the idea that mass does not affect the rate of acceleration in free fall, might predict no difference in the rate at which the objects roll down an inclined plane. Allow both cans to begin rolling at the same instant. Since the two cylinders have different moments of inertia, the can full of sand will accelerate at a greater rate and reach the bottom first (with a higher velocity). This provides students with an interesting anomaly about motion.