3.4 Equations of Uniformly Accelerated Motion | Translatory Motion | Physics 11

The equations of motion are a set of fundamental formulas in kinematics that describe the relationship between displacement, velocity, acceleration, and time for an object moving with constant acceleration in a straight line. These equations allow us to predict the state of an object's motion at any given time, provided the acceleration is uniform.

The five variables involved are:

s: displacement
vᵢ: initial velocity
vƒ: final velocity
a: constant acceleration
t: time interval

Derivation of the Equations

The three main equations of motion can be derived graphically from a velocity-time (v-t) graph. For an object with uniform acceleration, the v-t graph is a straight line.

The slope of the v-t graph represents acceleration.
The area under the v-t graph represents displacement.

Consider an object that accelerates from an initial velocity $v_{i}$ to a final velocity $v_{f}$ over a time interval $t$ .

1. First Equation of Motion: Velocity-Time Relation

This equation relates the final velocity of an object to its initial velocity, acceleration, and the time elapsed.

Formula: $v_{f} = v_{i} + a t$

Derivation: Acceleration (a) is defined as the slope of the velocity-time graph. $Slope = a = \frac{Change in Velocity}{Change in Time} = \frac{v _{f} - v _{i}}{t}$ Rearranging this equation to solve for $v_{f}$ , we get: $a t = v_{f} - v_{i}$ $v_{f} = v_{i} + a t$

2. Second Equation of Motion: Displacement-Time Relation

This equation calculates the displacement of an object given its initial velocity, acceleration, and the time interval.

Formula: $s = v_{i} t + \frac{1}{2} a t^{2}$

Derivation: Displacement (s) is the total area under the v-t graph. This area can be seen as a trapezium. The area of a trapezium is given by: $s = Area = \frac{1}{2} \times (sum of parallel sides) \times (height)$ In our graph, the parallel sides are the initial velocity ( $v_{i}$ ) and final velocity ( $v_{f}$ ), and the height is time (t). $s = \frac{1}{2} (v_{i} + v_{f}) t$ Now, we can substitute the expression for $v_{f}$ from the first equation of motion ( $v_{f} = v_{i} + a t$ ) into this area formula: $s = \frac{1}{2} (v_{i} + (v_{i} + a t)) t$ $s = \frac{1}{2} (2 v_{i} + a t) t$ Distributing the t gives the final form: $s = v_{i} t + \frac{1}{2} a t^{2}$

3. Third Equation of Motion: Velocity-Displacement Relation

Velocity-Time Graph for Uniform Acceleration

This equation relates the final and initial velocities to acceleration and displacement, notably without requiring time.

Formula: $2 a s = v_{f}^{2} - v_{i}^{2}$

Derivation: We start again with the formula for displacement as the area under the graph: $s = \frac{1}{2} (v_{i} + v_{f}) t$ From the first equation of motion, we can express time as $t = \frac{v _{f} - v _{i}}{a}$ . Substituting this expression for t into the displacement formula: $s = \frac{1}{2} (v_{f} + v_{i}) (\frac{v _{f} - v _{i}}{a})$ Multiplying the terms in the parentheses gives a difference of squares: $s = \frac{v _{f}^{2} - v _{i}^{2}}{2 a}$ Rearranging this gives the final form: $2 a s = v_{f}^{2} - v_{i}^{2}$

To ensure these equations are physically valid, they must satisfy the Principle of Homogeneity, meaning the Dimensions→ on both sides of the equation must be identical.

Summary

Equation Number	Formula	Variables Related	Variable Not Included
1st Equation	$v_{f} = v_{i} + a t$	Velocity, Acceleration, Time	Displacement (s)
2nd Equation	$s = v_{i} t + \frac{1}{2} a t^{2}$	Displacement, Velocity, Time	Final Velocity (vƒ)
3rd Equation	$2 a s = v_{f}^{2} - v_{i}^{2}$	Velocity, Acceleration, Displacement	Time (t)

These equations form the bedrock of kinematics and are essential for solving problems involving objects moving with constant acceleration, from a falling apple to a launching rocket.

References

Equations of Uniformly Accelerated Motion (FBISE Physics)

Derivation of the Equations

The three main equations of motion can be derived graphically from a velocity-time (v-t) graph. For an object with uniform acceleration, the v-t graph is a straight line.

The slope of the v-t graph represents acceleration.

The area under the v-t graph represents displacement.

Consider an object that accelerates from an initial velocity

v_{i}

to a final velocity

v_{f}

over a time interval

t

This equation relates the final velocity of an object to its initial velocity, acceleration, and the time elapsed.

Formula:

v_{f} = v_{i} + a t

Derivation: Acceleration (a) is defined as the slope of the velocity-time graph.

Slope = a = \frac{Change in Velocity}{Change in Time} = \frac{v _{f} - v _{i}}{t}

Rearranging this equation to solve for

v_{f}

, we get:

a t = v_{f} - v_{i}

v_{f} = v_{i} + a t

This equation calculates the displacement of an object given its initial velocity, acceleration, and the time interval.

Formula:

s = v_{i} t + \frac{1}{2} a t^{2}

Derivation: Displacement (s) is the total area under the v-t graph. This area can be seen as a trapezium. The area of a trapezium is given by:

s = Area = \frac{1}{2} \times (sum of parallel sides) \times (height)

In our graph, the parallel sides are the initial velocity (

v_{i}

) and final velocity (

v_{f}

), and the height is time (t).

s = \frac{1}{2} (v_{i} + v_{f}) t

Now, we can substitute the expression for

v_{f}

from the first equation of motion (

v_{f} = v_{i} + a t

) into this area formula:

s = \frac{1}{2} (v_{i} + (v_{i} + a t)) t

s = \frac{1}{2} (2 v_{i} + a t) t

Distributing the t gives the final form:

s = v_{i} t + \frac{1}{2} a t^{2}

This equation relates the final and initial velocities to acceleration and displacement, notably without requiring time.

Formula:

2 a s = v_{f}^{2} - v_{i}^{2}

Derivation: We start again with the formula for displacement as the area under the graph:

s = \frac{1}{2} (v_{i} + v_{f}) t

From the first equation of motion, we can express time as

t = \frac{v _{f} - v _{i}}{a}

. Substituting this expression for t into the displacement formula:

s = \frac{1}{2} (v_{f} + v_{i}) (\frac{v _{f} - v _{i}}{a})

Multiplying the terms in the parentheses gives a difference of squares:

s = \frac{v _{f}^{2} - v _{i}^{2}}{2 a}

Rearranging this gives the final form:

2 a s = v_{f}^{2} - v_{i}^{2}

To ensure these equations are physically valid, they must satisfy the Principle of Homogeneity, meaning the Dimensions→ on both sides of the equation must be identical.

Summary

Equation Number	Formula	Variables Related	Variable Not Included
1st Equation	$v_{f} = v_{i} + a t$	Velocity, Acceleration, Time	Displacement (s)
2nd Equation	$s = v_{i} t + \frac{1}{2} a t^{2}$	Displacement, Velocity, Time	Final Velocity (vƒ)
3rd Equation	$2 a s = v_{f}^{2} - v_{i}^{2}$	Velocity, Acceleration, Displacement	Time (t)

These equations form the bedrock of kinematics and are essential for solving problems involving objects moving with constant acceleration, from a falling apple to a launching rocket.