Table of Contents
Acceleration and Equations of Motion
1. Introduction
Acceleration describes how quickly the velocity of an object changes with time.
A moving vehicle accelerates when it increases its speed, decelerates when its speed decreases, and also accelerates when it changes direction, even if its speed remains constant.
Acceleration is a fundamental concept in mechanics because it connects motion with force through Newton’s second law.
The study of acceleration and equations of motion is essential for solving problems involving vehicles, falling objects, projectiles and other moving bodies.
2. Meaning of Acceleration
Acceleration is the rate of change of velocity with respect to time.
Average acceleration is:
[
a=\frac{v-u}{t}
]
Where:
- (a) = average acceleration
- (u) = initial velocity
- (v) = final velocity
- (t) = elapsed time
The SI unit of acceleration is metre per second squared (m/s²).
Acceleration is a vector quantity because it has both magnitude and direction.
2.1 Example
A car increases its velocity from 10 m/s to 30 m/s in 5 seconds.
[
a=\frac{30-10}{5}
]
[
a=4\ m/s^2
]
The car’s velocity increases by 4 m/s every second, assuming constant acceleration.
3. Types of Acceleration
3.1 Uniform acceleration
Uniform acceleration occurs when velocity changes by equal amounts in equal intervals of time.
For example, if a vehicle’s velocity increases by 5 m/s every second, its acceleration is uniform.
3.2 Non-uniform acceleration
Non-uniform acceleration occurs when velocity changes by unequal amounts in equal intervals of time.
This is common in real-world traffic conditions.
3.3 Positive acceleration
Positive acceleration means acceleration acts in the positive direction of the chosen coordinate system.
It does not necessarily mean that an object is speeding up. The effect depends on the direction of velocity.
3.4 Negative acceleration
Negative acceleration means acceleration acts in the negative direction of the chosen coordinate system.
It is not always equivalent to deceleration. An object moving in the negative direction can speed up under negative acceleration.
3.5 Retardation or deceleration
Retardation refers to acceleration opposite to the direction of velocity, causing the speed of an object to decrease.
For example, a car braking while moving forward experiences acceleration opposite to its velocity.
4. Acceleration in Different Situations
| Situation | Velocity | Acceleration | Effect |
|---|---|---|---|
| Car speeding up forward | Positive | Positive | Speed increases |
| Car slowing down forward | Positive | Negative | Speed decreases |
| Object speeding up backward | Negative | Negative | Speed increases |
| Object slowing down backward | Negative | Positive | Speed decreases |
| Object moving at constant velocity | Constant | Zero | No change in velocity |
| Object moving in a circle at constant speed | Changing direction | Non-zero | Direction changes |
5. Equations of Motion
The equations of motion describe the relationship between displacement, initial velocity, final velocity, acceleration and time for an object moving with constant acceleration in a straight line.
These are commonly known as the three equations of uniformly accelerated motion.
The equations assume:
- Motion along a straight line.
- Constant acceleration.
- A consistent sign convention.
- Time measured from the beginning of the interval under consideration.
5.1 First equation of motion
[
v=u+at
]
Where:
- (v) = final velocity
- (u) = initial velocity
- (a) = constant acceleration
- (t) = time
This equation is used when initial velocity, acceleration and time are known or when one of these quantities must be calculated.
Example:
A vehicle starts with a velocity of 5 m/s and accelerates uniformly at 3 m/s² for 4 seconds.
[
v=5+(3\times4)
]
[
v=17\ m/s
]
The final velocity is 17 m/s.
5.2 Second equation of motion
[
s=ut+\frac{1}{2}at^2
]
Where:
- (s) = displacement
- (u) = initial velocity
- (a) = constant acceleration
- (t) = time
This equation calculates displacement when initial velocity, acceleration and time are known.
Example:
A vehicle starts with an initial velocity of 4 m/s and accelerates at 2 m/s² for 5 seconds.
[
s=(4\times5)+\frac{1}{2}(2)(5^2)
]
[
s=20+25
]
[
s=45\ m
]
The displacement is 45 metres.
5.3 Third equation of motion
[
v^2=u^2+2as
]
This equation relates initial velocity, final velocity, acceleration and displacement without requiring time.
Example:
A vehicle moving at 10 m/s accelerates uniformly at 2 m/s² over a distance of 50 metres.
[
v^2=10^2+2(2)(50)
]
[
v^2=100+200=300
]
[
v=\sqrt{300}\approx17.32\ m/s
]
The final velocity is approximately 17.32 m/s, assuming the direction remains positive.
5.4 Average velocity under constant acceleration
For constant acceleration in one-dimensional motion:
[
v_{\text{avg}}=\frac{u+v}{2}
]
Therefore:
[
s=\frac{u+v}{2}t
]
This relation is valid when acceleration is constant.
6. Derivation of the Equations of Motion
6.1 Derivation of the first equation
Acceleration is the change in velocity per unit time:
[
a=\frac{v-u}{t}
]
Multiplying both sides by time:
[
at=v-u
]
Rearranging:
[
v=u+at
]
This is the first equation of motion.
6.2 Derivation of the second equation
For constant acceleration, average velocity is:
[
v_{\text{avg}}=\frac{u+v}{2}
]
Displacement is average velocity multiplied by time:
[
s=\frac{u+v}{2}t
]
Using the first equation:
[
v=u+at
]
Substituting:
[
s=\frac{u+(u+at)}{2}t
]
[
s=\frac{2u+at}{2}t
]
Therefore:
[
s=ut+\frac{1}{2}at^2
]
6.3 Derivation of the third equation
From the first equation:
[
v-u=at
]
From the second equation:
[
s=\frac{u+v}{2}t
]
Substituting (t=(v-u)/a):
[
s=\frac{u+v}{2}\times\frac{v-u}{a}
]
[
2as=(u+v)(v-u)
]
Using the difference of two squares:
[
2as=v^2-u^2
]
Therefore:
[
v^2=u^2+2as
]
7. Free Fall and Acceleration Due to Gravity
Free fall is the motion of an object under the influence of gravity alone.
In ideal free fall, air resistance is neglected.
Near Earth’s surface, the acceleration due to gravity is approximately:
[
g=9.8\ m/s^2
]
The more precise conventional standard value is 9.80665 m/s², although actual local gravitational acceleration varies with latitude, altitude and local geological conditions.
7.1 Object falling from rest
For an object released from rest:
- Initial velocity (u=0)
- Acceleration (a=g), if downward is positive
- Displacement is measured downward
The equations become:
[
v=gt
]
[
s=\frac{1}{2}gt^2
]
[
v^2=2gs
]
Example:
An object is dropped from rest and falls freely for 3 seconds. Calculate its final velocity and distance travelled, taking (g=9.8\ m/s^2).
Final velocity:
[
v=0+(9.8\times3)=29.4\ m/s
]
Distance:
[
s=\frac{1}{2}(9.8)(3^2)
]
[
s=44.1\ m
]
The object reaches a speed of 29.4 m/s after falling 44.1 metres.
7.2 Object thrown vertically upward
When an object is thrown vertically upward, gravity acts downward and reduces its upward velocity.
Taking upward as positive:
[
a=-g
]
The equations become:
[
v=u-gt
]
[
s=ut-\frac{1}{2}gt^2
]
[
v^2=u^2-2gs
]
At the highest point, instantaneous vertical velocity becomes zero.
The time to reach maximum height is:
[
t=\frac{u}{g}
]
The maximum height above the point of projection is:
[
H=\frac{u^2}{2g}
]
If the object returns to its original level and air resistance is neglected, its total time of flight is:
[
T=\frac{2u}{g}
]
8. Motion Graphs and Acceleration
8.1 Velocity-time graph
The gradient of a velocity-time graph represents acceleration.
[
a=\frac{\text{Change in Velocity}}{\text{Time}}
]
A horizontal velocity-time line indicates constant velocity and zero acceleration.
A straight sloping line indicates constant acceleration.
A curved line indicates changing acceleration or a changing rate of velocity.
8.2 Displacement-time graph
The gradient of a displacement-time graph represents velocity.
A straight line with constant gradient indicates constant velocity.
A changing gradient indicates changing velocity.
8.3 Acceleration-time graph
The area under an acceleration-time graph represents the change in velocity.
For constant acceleration, the graph is a horizontal line.
9. Uniform Circular Motion
Uniform circular motion occurs when an object moves along a circular path at constant speed.
Although speed remains constant, velocity changes because direction changes continuously.
Therefore, the object experiences centripetal acceleration directed towards the centre of the circle.
[
a_c=\frac{v^2}{r}
]
Where:
- (v) = tangential speed
- (r) = radius
The time required to complete one revolution is called the period, (T).
The frequency is the number of revolutions per second.
[
f=\frac{1}{T}
]
Angular velocity is:
[
\omega=\frac{2\pi}{T}=2\pi f
]
Tangential speed is related to angular velocity by:
[
v=r\omega
]
10. Limitations of the Equations of Motion
The three standard equations are applicable to straight-line motion with constant acceleration.
They should not be applied directly when:
- Acceleration varies significantly with time.
- Motion is curved and vector directions must be considered.
- Air resistance or other forces cause non-constant acceleration.
- The problem involves relativistic speeds requiring relativistic mechanics.
For variable acceleration, calculus-based methods and appropriate motion graphs may be needed.
36 Key Takeaways for Exams
- Acceleration is the rate of change of velocity with time.
- Acceleration is a vector quantity.
- The SI unit of acceleration is m/s².
- Average acceleration equals change in velocity divided by elapsed time.
- Uniform acceleration produces equal changes in velocity in equal time intervals.
- Non-uniform acceleration involves unequal changes in velocity over equal time intervals.
- Positive acceleration does not always mean an object is speeding up.
- Negative acceleration does not always mean an object is slowing down.
- Retardation occurs when acceleration acts opposite to velocity.
- Constant velocity implies zero acceleration.
- A change in direction produces acceleration even when speed is constant.
- The three equations of motion apply to straight-line motion with constant acceleration.
- The first equation of motion relates velocity, acceleration and time.
- The second equation of motion relates displacement, initial velocity, acceleration and time.
- The third equation of motion relates velocity, acceleration and displacement without time.
- Average velocity under constant acceleration is the arithmetic mean of initial and final velocities.
- The gradient of a velocity-time graph represents acceleration.
- The gradient of a displacement-time graph represents velocity.
- The signed area under a velocity-time graph represents displacement.
- The area under an acceleration-time graph represents change in velocity.
- Free fall is motion under gravity alone when air resistance is neglected.
- Near Earth’s surface, gravitational acceleration is approximately 9.8 m/s².
- A freely falling object released from rest has zero initial velocity.
- A freely falling object gains approximately 9.8 m/s of downward velocity each second near Earth’s surface.
- At the highest point of vertical upward motion, instantaneous velocity is zero.
- Gravitational acceleration continues to act downward at the highest point.
- The maximum height of a vertically projected object depends on the square of its initial velocity.
- In ideal vertical projection, time of ascent equals time of descent to the same level.
- Uniform circular motion has constant speed but changing velocity.
- Centripetal acceleration is directed towards the centre of a circular path.
- The magnitude of centripetal acceleration increases with the square of speed.
- The period is the time required for one complete cycle or revolution.
- Frequency is the number of cycles or revolutions per second.
- Frequency and period are reciprocals.
- Angular velocity is measured in radians per second.
- Standard equations of motion cannot be used without modification for variable acceleration.