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Acceleration
- Define and describe acceleration.
- Describe and analyze motion in one dimension using equations for acceleration.
- Determine acceleration from the slope of the velocity versus time graph.
Objectives
Assessment
- Define and describe accelerated motion.
- A car changes its velocity from 0 to 20 m/s in 4.0 seconds. What is its acceleration?
- A car initially traveling at 25 m/s comes to a stop in 3.0 s. What is its acceleration?
- An object starts from rest and accelerates at 2.5 m/s^2 for 10 seconds. What is its final velocity? 5. The motion of a particle along a straight line is depicted in this graph.
Assessment
V
(m/s) t (s) What is the acceleration of the particle from 7 seconds to 10 seconds?
Velocity vs. time
Physics terms
- velocity
- acceleration
- rate of change
Equations
Acceleration is the change in velocity divided by the change in time.
Brainstorm: An accelerating object has a changing velocity. Can you share an example of something accelerating? a coaster making a turn Examples a coaster making a turn a ball rolling uphill Examples a coaster making a turn a ball rolling uphill a baseball falling Examples 0 m/s 4 m/s 4 m/s 4 m/s 4 m/s 4 m/s 2 m/s Two balls are moving to the right. Their velocity at each second is shown.
- Which ball is accelerating?
- What is its acceleration? 2 6 m/s What is acceleration? 0 m/s 4 m/s 4 m/s 4 m/s 4 m/s 4 m/s 2 m/s Two balls are moving to the right. Their velocity at each second is shown.
- Which ball is accelerating?
- What is its acceleration? Ball 2 is accelerating at +2 m/s per second:
6 m/s a = +2 m/s 2
What is acceleration? 2
If an object has an acceleration of -4 m/s^2 , its velocity changes by -4 m/s each second. For example, if it has an initial velocity of +8 m/s and accelerates for one second... Understanding the concept
vi
(initial velocity)
vf
(final velocity)
Δ v
(change in velocity) If an object has an acceleration of -4 m/s^2 , its velocity changes by -4 m/s each second. For example, if it has an initial velocity of +8 m/s and accelerates for one second, its vf is +4 m/s. Understanding the concept Object 1 Ball 2 What is the acceleration of object 1? The meaning of acceleration This chart shows the position and speed of an object moving to the right for three consecutive seconds. 0 m/s 1.5 m/s 3 m/s 4.5 m/s 2 m/s 6 m/s 10 m/s Object 1 Ball 2Object 2 What is the acceleration of object 1? +1.5 m/s^2 What is the acceleration of object 2? The meaning of acceleration This chart shows the position and speed of an object moving to the right for three consecutive seconds. 0 m/s 1.5 m/s 3 m/s 4.5 m/s 2 m/s 6 m/s 10 m/s? Object 1 Ball 2Object 2 What is the acceleration of object 1? +1.5 m/s^2 What is the acceleration of object 2? +4 m/s^2 What is the missing velocity? The meaning of acceleration This chart shows the position and speed of an object moving to the right for three consecutive seconds. 0 m/s 1.5 m/s 3 m/s 4.5 m/s 2 m/s 6 m/s 10 m/s 14 m/s Object 1 Ball 2Object 2 What is the acceleration of object 1? +1.5 m/s^2 What is the acceleration of object 2? +4 m/s^2 What is the missing velocity? +14 m/s The meaning of acceleration This chart shows the position and speed of an object moving to the right for three consecutive seconds. 0 m/s 1.5 m/s 3 m/s 4.5 m/s
Click on this interactive calculator on page 110. Exploring the ideas A car starts at rest and accelerates to a velocity of 10 m/s in 5 seconds. What is its acceleration? Enter the values and click [Run] to see the car perform the motion. Engaging with the concepts Acceleration 10 5 10 0 5 0 A car starts at rest and accelerates to a velocity of 10 m/s in 5 seconds. What is its acceleration?
a = 2 m/s^2
Engaging with the concepts Acceleration 10 5 10 0 5 0 2 A race car accelerates from 0 m/s to 27 m/s (60 mph) in 4 seconds. What is its acceleration? Engaging with the concepts Acceleration 27 4 27 0 4 0 Engaging with the concepts Acceleration 27 4 27 0 4 0
a = 6.75 m/s^2 6.
A race car accelerates from 0 m/s to 27 m/s (60 mph) in 4 seconds. What is its acceleration? Engaging with the concepts Acceleration 0 0 0 0 0 0 Can you make the car go forward and then backward? How?
Between 0 and 2 seconds the acceleration is +2 m/s^2. Between 2 and 3 seconds the acceleration is zero. Between 3 and 6 seconds the acceleration is -3 m/s^2.
What is happening to the speed?
The meaning of acceleration decreases increases constant
Speed
(m/s)
Positive AND negative acceleration can cause speed to increase OR decrease – depending on the direction of the velocity! increases
Between 0 and 2 seconds the acceleration is +2 m/s^2. Between 2 and 3 seconds the acceleration is zero. Between 3 and 6 seconds the acceleration is -3 m/s^2. The meaning of acceleration Positive acceleration of +4 m/s^2 adds +4 m/s of velocity each second. 0 m/s 4 m/s 8 m/s 12 m/s 16 m/s Signs of the acceleration Positive acceleration of +4 m/s^2 adds +4 m/s of velocity each second. Negative acceleration of -4 m/s^2 adds -4 m/s of velocity each second. 0 m/s 4 m/s 8 m/s 12 m/s 16 m/s 16 m/s 12 m/s 8 m/s 4 m/s 0 m/s Signs of the acceleration What is the sign of acceleration in each of these four possible cases? 0 m/s +1 m/s +2 m/s + 3 m/s Signs of the acceleration
- speeding up in + direction:?
- slowing down in + direction:? 0 m/s +1 m/s +2 m/s + 3 m/s +3 m/s^ +2 m/s^ +1 m/s^ 0 m/s
Signs of the acceleration
What is the sign of acceleration in each of these four possible cases?
1. speeding up in + direction: + a
0 m/s +1 m/s +2 m/s + 3 m/s +3 m/s^ +2 m/s^ +1 m/s^ 0 m/s -3 m/s -2 m/s -1 m/s 0 m/s
Signs of the acceleration
- speeding up in - direction:? What is the sign of acceleration in each of these four possible cases?
1. speeding up in + direction: + a 2. slowing down in + direction: - a
0 m/s +1 m/s +2 m/s + 3 m/s +3 m/s^ +2 m/s^ +1 m/s^ 0 m/s -3 m/s -2 m/s -1 m/s 0 m/s 0 m/s -1 m/s -2 m/s -3 m/s
Signs of the acceleration
What is the sign of acceleration in each of these four possible cases?
1. speeding up in + direction: + a 2. slowing down in + direction: - a
3. speeding up in - direction: - a 4. slowing down in - direction:?
1. speeding up in + direction: + a
4. slowing down in - direction: + a!
0 m/s +1 m/s +2 m/s + 3 m/s +3 m/s +2 m/s +1 m/s 0 m/s -3 m/s -2 m/s -1 m/s 0 m/s 0 m/s -1 m/s -2 m/s -3 m/s
2. slowing down in + direction: - a
Signs of the acceleration
3. speeding up in - direction: - a
What is the sign of acceleration in each of these four possible cases? Test your knowledge A car is headed west (the negative direction) on a long straight road. The driver sees a red light up ahead and slows to a stop. Is the car’s acceleration positive or negative? Test your knowledge A car is headed west (the negative direction) on a long straight road. The driver sees a red light up ahead and slows to a stop. Is the car’s acceleration positive or negative? Slowing down in the
negative direction is + a!
Test your knowledge What does -1 m/s^2 mean? Can you describe two ways that an object could have a negative acceleration?
acceleration^ No constant velocity acceleration^ No constant velocity acceleration^ Positive changing velocity
- A car moves at a constant speed of 3 m/s for 3 seconds.
- The car accelerates to 6 m/s over the next 3 seconds.
- The car continues at 6 m/s for three more seconds. What does this look like? Acceleration on the v vs. t graph acceleration^ Positive changing velocity Acceleration causes the velocity vs. time graph to have a non-zero slope. Acceleration on the v vs. t graph Acceleration on the v vs. t graph Calculate the acceleration from t = 3 to t = 6 seconds. Δ v Δ t Acceleration on the v vs. t graph 6 m/s 3 m/s 3 s +1 m/s^2 Δ v Δ t Calculate the acceleration from t = 3 to t = 6 seconds. Acceleration on the v vs. t graph (^) An object starting from rest accelerates at 1 m/s^2. Its velocity increases with time, making a linear v vs. t graph. What does the position vs. time graph look like? Position vs. time
An object starting from rest accelerates at 1 m/s^2. Its velocity increases with time, making a linear v vs. t graph. What does the position vs. time graph look like?
- As the velocity increases the slope must change! Position vs. time An object starting from rest accelerates at 1 m/s^2. Its velocity increases with time, making a linear v vs. t graph. What does the position vs. time graph look like?
- As the velocity increases the slope must change!
The graph is a curve.
Position vs. time Acceleration creates a sloped line on a v vs. t graph. Acceleration creates a curve on an x vs. t graph. Curves vs. lines The x vs. t graph curves
upward for + a , like a smile!
The x vs. t graph curves
downward for - a , like a frown!
Curves vs. lines
- How does statement (A) relate to the diagram below it? What does “± 5 mph/s” mean?
- What does the word “rate” mean in the context of statement (B)? How do the arrows represent the idea of a rate of changing velocity? Understanding acceleration Understanding acceleration
- How does the equation represent statement (C)? Translate the symbols and operations into English. Give numerical examples of a change in velocity divided by a change in time.
Assessment A change of -25 m/s over 3 seconds is an acceleration of -8.3 m/s^2.
- A car initially traveling at 25 m/s comes to a stop in 3.0 s. What is its acceleration? Assessment 4. An object starts from rest and accelerates at 2.0 m/s^2 for 10 seconds. What is its final velocity? Assessment
- An object starts from rest and accelerates at 2.0 m/s^2 for 10 seconds. What is its final velocity? The final velocity is 20 m/s. 0 s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 s 9 s 10 s 0 m/s 2 m/s 4 m/s 6 m/s 8 m/s 10 m/s 12 m/s 14 m/s 16 m/s 18 m/s 20 m/s Assessment
- The motion of a particle along a straight line is depicted in this graph. What is the acceleration of the particle from 7 seconds to 10 seconds?
Velocity vs. time
V
(m/s) t (s)
- The motion of a particle along a straight line is depicted in this graph. Assessment answer: 40 m/s^2 What is the acceleration of the particle from 7 seconds to 10 seconds? t (s)
Velocity vs. time
V
(m/s)
