Describing Motion Around Us · Lesson 4 of 10
Average acceleration
“Acceleration is what turns a calm bus ride into an unexpected balance test.”
• Calculate average acceleration from initial and final velocities. • Interpret the sign and direction of acceleration. • Distinguish high velocity from high acceleration. • Explain acceleration caused by magnitude or direction changes. • Recognise constant acceleration from equal velocity changes. • Use a consistent sign convention in vertical motion.
When a bus begins moving, passengers may lean backward; when it brakes, they may lean forward. These experiences do not tell us merely that the bus is fast or slow. They reveal that its velocity is changing. Acceleration measures how quickly that change occurs.
Average acceleration is the change in velocity divided by the time interval during which the change occurs.
| Symbol | Meaning | Unit |
|---|---|---|
| u | Initial velocity | m s⁻¹ |
| v | Final velocity | m s⁻¹ |
| t₂−t₁ | Time interval | s |
| a | Average acceleration | m s⁻² |
If velocity increases in the chosen positive direction, acceleration is positive and points with the velocity. If the magnitude of velocity decreases while the object continues in the positive direction, acceleration points oppositely and is negative. The sign describes direction relative to the selected convention; it is not a label meaning “good” or “bad.”
Problem
A bus changes from 36 km h⁻¹ to 54 km h⁻¹ in 10 s, then later changes from 54 km h⁻¹ to rest in 5 s. Find both accelerations.
- 1.Convert 36 km h⁻¹: (36×5/18=10 m s⁻¹).
- 2.Convert 54 km h⁻¹: (54×5/18=15 m s⁻¹).
- 3.While accelerating, (a=(15-10)/10=0.5 m s⁻²).
- 4.While braking, (u=15 m s⁻¹), (v=0) and (t=5 s).
- 5.Therefore, (a=(0-15)/5=-3 m s⁻²).
- 6.The negative sign shows that braking acceleration is opposite to the positive direction of motion.
Activity: Let us calculate
Vehicle performance is sometimes described by the time needed to change from 0 km h⁻¹ to 100 km h⁻¹. To compare vehicles fairly, convert 100 km h⁻¹ to metres per second and divide by each recorded time. A shorter time for the same velocity change means a greater acceleration magnitude.
Problem
A car reaches 100 km h⁻¹ from rest in 8 s. Find its average acceleration magnitude.
- 1.Convert the final velocity: (100×5/18≈27.78 m s⁻¹).
- 2.Initial velocity (u=0 m s⁻¹).
- 3.Time interval (t=8 s).
- 4.Use (a=(v-u)/t).
- 5.(a=(27.78-0)/8≈3.47 m s⁻²).
Acceleration is constant when velocity changes by equal amounts in equal intervals of time.
Problem
Velocities at successive seconds are 0, 9.8, 19.6, 29.4 and 39.2 m s⁻¹ downward. Show that acceleration is constant.
- 1.Choose downward as the positive direction.
- 2.From 0 s to 1 s: (a=(9.8-0)/1=9.8 m s⁻²).
- 3.From 1 s to 2 s: (a=(19.6-9.8)/1=9.8 m s⁻²).
- 4.The same subtraction for later intervals also gives (9.8 m s⁻²).
- 5.Equal velocity changes in equal time intervals show constant acceleration.
An object can move very fast and still have zero acceleration if its velocity remains unchanged. Conversely, an object moving at constant speed along a curve has acceleration because the direction of velocity changes. Acceleration can result from a change in magnitude, direction or both.
Ready to Go Beyond
Acceleration at one instant is called instantaneous acceleration. Like instantaneous velocity, it can be approached by examining changes over progressively smaller time intervals. The present treatment focuses on average acceleration and cases where acceleration is constant.
Note
The origin and positive direction may be chosen for convenience. In a falling-object problem, downward can be positive; in an upward-throw problem, upward may be positive. Either convention works if every velocity, displacement and acceleration is assigned a consistent sign.
Whether speed increases or decreases depends on the relative directions of velocity and acceleration. Opposite directions reduce speed; the same direction increases it.
Quiz
Which description best matches Average Acceleration?
Which description best matches Constant Acceleration?
Which term matches this description: Average acceleration is the change in velocity divided by the time interval during which the change occurs.
Which term matches this description: Acceleration is constant when velocity changes by equal amounts in equal intervals of time.
Which statement is a key takeaway from this lesson?
Practice Problems
- A scooter changes from 5 m s⁻¹ to 17 m s⁻¹ in 4 s. Find acceleration.
- A car travelling at 20 m s⁻¹ stops in 8 s. Find its acceleration using forward as positive.
- Explain how an object can have high velocity and zero acceleration.
- Explain how acceleration can occur at constant speed.
Key Takeaways
• Acceleration is the rate of change of velocity. • Its direction depends on the chosen sign convention. • Opposite velocity and acceleration directions reduce speed. • Constant velocity means zero acceleration. • Direction change alone can produce acceleration.