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List of Sources:

1. Young et al., 2016¹
2. Photo by cottonbro studio on Pexels.com
3. Graphs generated using Desmos.com

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Table of Contents:

1. Displacement, Time and Average Velocity (Young et al., 2016)
   1. Displacement
   2. Time
   3. Average Velocity
   4. Graphical Representation
2. Instantaneous Velocity (Young et al., 2016)
   1. Graphical Representation
   2. The Slope
   3. Motion Diagram
3. Average and Instantaneous Acceleration (Young et al., 2016)
   1. Average Acceleration
      1. Graphical Representation
   2. Instantaneous Acceleration
      1. Graphical Representation
      2. Motion Diagram
4. Motion with Constant Acceleration (Young et al., 2016)
   1. Equations of Motion with Constant Acceleration
5. Free Falling Bodies (Young et al., 2016)
6. Motion when Acceleration is not Constant (Young et al., 2016)
   1. Velocity and Acceleration by differentiation
   2. Velocity and Position by Integration
7. Common Errors in Understanding (Young et al., 2016)

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Displacement, Time and Average Velocity (Young et al., 2016)

For the purposes of understanding motion, macroscopic objects are often treated as particles.

Displacement

Consider a particle in 1D located at point P₁ along the x-axis at time t₁. After some time, the particle is found to be at point P₂ at time t₂. The component of the displacement vector along the x-direction is given by the change in the particle’s position which is equal to the difference between final position, given by the x-coordinate x₂, and the initial position, given by the x-coordinate x₁.

In other words, component of displacement along the x-axis is given by;

Δx² = x₂ – x₁……(1)

The change in position occurs only along the x-axis which means that the y and z components of the displacement vector will be equal to zero, i.e., Δy = Δz = 0.

The direction of the displacement vector is towards the positive x-axis as depicted by the vector in Figure 1. In other words, if the particle’s initial position is positive and it starts to become more positive, we can conclude that it is moving towards the positive x-direction.

Time

The time interval is defined as the change in time as the particle displaces from point P₁ to P₂. This can be written as,

Δt = t₂ – t₁……(2)

Average Velocity

The average x-velocity (or component of average velocity along the x-direction) is then given as the component of the displacement vector along the x-axis divided by the time interval;

…..(3)

If x is given in meters (m) and t in seconds (s), Δx/Δt has units of m/s which correctly correspond to the units of velocity. The direction of average x-velocity of the particle in Figure 1 is positive³ since the displacement vector is positive⁴.

Scenario 1 is illustrated in Figure 1. The other three scenarios are shown below:

Scenario 2:

Scenario 3:

Scenario 4:

Graphical Representation

Just like displacement is independent of the path taken by the particle, average velocity is independent of the particle’s velocity at any instant between points P₁ and P₂.

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Instantaneous Velocity (Young et al., 2016)

If in Figure 1, we start moving P₂ closer to P₁ such that Δx and Δt become infinitesimally small, the instantaneous velocity is given by the average x-velocity in the limit that Δt approaches zero, which is also called the derivative of x with respect to t.

…….(4)

The derivative of x with respect to t is described as the instantaneous rate of change in particle’s position. Similar to average x-velocity, the direction of instantaneous velocity is the same as the direction of the displacement vector⁶ (See Error #1).

Note: Instantaneous velocity (not average velocity) is what is meant when the word ‘velocity’ is used by itself.

Graphical Representation

The instantaneous velocity of a particle, v_x can be determined from the x-t graph.

As the second point starts to approach the first one on x-t curve, the slope of the line between the two points approaches the slope of the tangent to the curve drawn at the first point (see below).

Thus, the instantaneous x-velocity at a point, say P₁ on the x-t graph is equal to the slope of the tangent to the curve drawn at that point.

The Slope

The direction of the slope, whether it is positive or negative gives information about the direction of instantaneous velocity. If the slope is inclined upwards on the right (positive slope), velocity is in the positive x-direction. Conversely, if the slope is inclined downwards on the right (negative slope), velocity is in the negative x-direction.

The steepness of the slope sheds light on the magnitude of the instantaneous velocity. Steeper the slope, faster is the particle moving.

Position	Position along x-axis7	Slope	Particle’s position getting...	vx (direction)	vx (comparative magnitude)
A	On positive x-axis	Positive	…more positive	Positive x-direction	Not too steep, particle is picking up speed
B	On positive x-axis	Positive	…more positive	Positive x-direction	Much steeper, particle traveling much faster than position A
C	On positive x-axis	Zero	Momentarily at rest	N/A	Particle reached a maximum on the curve* and momentarily vx = 0
D	On positive x-axis	Negative	…less positive	Negative x-direction	Very steep, particle moving very fast
E	At x=0	Negative	Switching from positive to negative	Negative x-direction	Quite steep, but particle not as fast as at position D
F	On negative x-axis	Negative	…more negative	Negative x-direction	Steepness significantly dropped compared to position E, particle is slowing down
G	On negative x-axis	Zero	Momentarily at rest	N/A	Particle reached a minimum on the curve* and momentarily vx = 0
H	On negative x-axis	Positive	…less negative	Positive x-direction	Getting steep again as particle picks up speed

*Maximum and minimum on a curve simply describe the highest and lowest points on the curve indicative of the function flipping its direction. They do not correspond to maximum or minimum velocity of the particle. Velocity of the particle at both points is zero. At position C, the particle is farthest from origin along the positive x-axis and at position G, it is farthest from origin along the negative x- axis.

Motion Diagram

A Motion Diagram is used to show the position of a particle at certain instants of time with arrows that indicate the direction of the instantaneous velocity at that instant. The size of the arrow is equivalent to the magnitude of the instantaneous velocity. Larger is the size of the arrow, greater is the speed.

When the position of the particle at each instant in the motion diagram is stacked up sequentially, it creates a movie showing the movement of the particle along the x-axis (see below).

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Average and Instantaneous Acceleration (Young et al., 2016)

In general, acceleration is a vector quantity that describes the rate of change of the velocity vector.

Average Acceleration

Consider a particle in 1D moving along the x-axis such that at time t₁, it is located at point P₁ and it’s velocity is given by v_1x. It continues to move along the x-axis and reaches a point P₂ at time t₂ where the particle’s velocity is given by v_2x. The change in particle’s velocity, Δv_x can be stated as follows:

Δv_x = v_2x – v_1x….(5)

The average x-acceleration (or component of average acceleration in the x-direction) of the particle between points P₁ and P₂ is equal to the change in particle’s velocity, Δv_x, divided by the time interval, , Δt = t₂ – t₁;

……(6)

If v is given in meters/second (m/s) and t in seconds (s), Δv_x/Δt has units of m/s² which correctly correspond to the units of acceleration.

Direction of Average Acceleration

Scenario 1: Figure 11: The initial velocity vector, v1x is in the positive x-direction. The final velocity vector, v2x is larger in magnitude and in positive x-direction. The initial velocity of the particle is positive and it became more positive with time. Hence, the average acceleration is positive and particle is speeding up in +x-direction.	Scenario 2: Figure 12: The initial velocity vector, v1x is in the positive x-direction. The final velocity vector, v2x is smaller in magnitude and in positive x-direction. The initial velocity of the particle is positive and it became less positive with time. Hence, the average acceleration is negative and the particle is slowing down in the +x-direction.
Scenario 3: Figure 13: The initial velocity vector, v1x is in the negative x-direction. The final velocity vector, v2x is larger in magnitude and in negative x-direction. The initial velocity of the particle is negative and it became more negative with time. Hence, the average acceleration is negative and particle is speeding up in -x-direction.	Scenario 4: Figure 14: The initial velocity vector, v1x is in the negative x-direction. The final velocity vector, v2x is smaller in magnitude and in negative x-direction. The initial velocity of the particle is negative and it became less negative with time. Hence, the average acceleration is positive and particle is slowing down in the -x-direction.

The direction of the average acceleration vector is along the direction of the initial velocity vector (v_1x) if the particle is speeding up between points P₁ and P₂, and it is opposite to the direction of the initial velocity vector (v_1x) if the particle is slowing down between points P₁ and P₂.

Why do we look at the direction of initial velocity vector when identifying whether the average acceleration is positive or negative?

Consider a particle at point P₁ moving along the x-axis with a positive x-velocity, v_1x. After some time, the particle is at point P₂ and the particle’s x-velocity is now negative. Since the particle’s initial x-velocity is positive and it became less positive, we can conclude that the average acceleration is negative and particle is now speeding up in the negative x-direction. At P₂, the acceleration is in the same direction as the particle’s velocity at P₂.

The particle’s initial velocity is in the positive x-direction while the acceleration is negative. The particle starts to slow down in the positive x-direction, momentarily comes to a stop and finally, the particle reverses its direction and starts speeding up along the negative x-direction (see below).

Graphical Representation

When a particle’s x-velocity is graphed with respect to time, the slope of the line between points P₁ and P₂ is equivalent to the average acceleration in the x-direction.

Scenario 1 (Positive Slope) Figure 17: At P1, velocity is positive and it becomes more positive with time. We can conclude that the average acceleration between points P1 and P2 is positive and particle is speeding up in the +x-direction. This can be confirmed by the direction of the slope which is positive.	Scenario 2 (Positive (Slope) Figure 18: At P1, velocity is negative and it becomes less negative with time. We can conclude that the average acceleration between points P1 and P2 is positive and particle slowed down in the -x-direction and then started speeding up in the +x-direction.
Scenario 3 (Negative Slope) Figure 19: At P1, velocity is positive and it becomes less positive with time. We can conclude that the average acceleration between points P1 and P2 is negative and particle slowed down in the +x-direction.	Scenario 4 (Negative Slope) Figure 20: At P1, velocity is negative and it becomes more negative with time. We can conclude that the average acceleration between points P1 and P2 is negative and particle speeding up in the -x-direction.

Instantaneous Acceleration

If in Figure 16, we start moving P₂ closer to P₁ such that Δv_x and Δt become infinitesimally small, the instantaneous x-acceleration is given by the average x-acceleration in the limit that Δt approaches zero, which is also called the derivative of v_x with respect to t.

……(7)

Remember, a_x is the component of instantaneous acceleration in the x direction (See Error #2).

Graphical Representation

V_x-t Graph

The graph of instantaneous velocity (v_x) with respect to time (t) is called a v_x-t graph.

On a v_x-t graph, the instantaneous x-acceleration at a point is equal to the slope of the tangent to the curve at that point.

The Slope

If the slope of the tangent is positive, the direction of instantaneous acceleration is positive. In the other words, the acceleration is towards the positive x-axis. Conversely, if the slope of the tangent is negative, the direction of instantaneous acceleration is negative or its towards the negative x-direction.

Figure 22: v_x-t curve is represented in blue with black lines for tangents to the curve at points given in pink. Click on the black triangles for more information⁸.

X-t Graph

Information about the acceleration can also be drawn from an x-t graph. This is because instantaneous acceleration (a_x) is equal to the second derivative of position (x) with respect to time (t),

…..(8)

The curvature of an x-t curve at a point gives information about acceleration at that point. If the function is curved up (or concave up), acceleration is positive. If instead, the function is curved down (or concave down), acceleration is negative. If the function at the point is straight with no curvature, acceleration is zero and velocity is constant. This is true for second derivative of any function. Greater is the curvature of the function, larger is the magnitude of acceleration⁹.

Figure 23: The x-t curve is shown in purple with tangents drawn with black at points given in pink. Click on the black triangles for more information.

Motion Diagram

A motion diagram, as discussed previously, can be created for Figure 23.

Motion with Constant Acceleration (Young et al., 2016)

A particle moving with constant acceleration means that the velocity of the particle is changing by an equal amount over each time interval which is held constant.

Consider a particle starting with a velocity, v_x = 2 m/s at t = 0 s. The time interval, Δt, is held constant and taken to be 1s. The subsequent changes in velocity are shown in the following table:

t	vx
0	2
1	4
2	6
3	8

For t₁ = 0 s and t₂ = 1 s, Δt = 1 s and Δv_x = (4 -2) m/s = 2 m/s

a_av(x) = Δv/Δt = 2 m/s²

For t₁ = 1 s and t₂ = 2 s, Δt = 1 s and Δv_x = (6 -4) m/s = 2 m/s

a_av(x) = Δv/Δt = 2 m/s²

For t₁ = 2 s and t₂ = 3 s, Δt = 1 s and Δv_x = (8 -6) m/s = 2 m/s

a_av(x) = Δv/Δt = 2 m/s²

Velocity over a time interval Δt = 1 s changes at a fixed rate giving an acceleration that is constant. In other words, velocity is increasing by 2 m/s every second.

Particle’s motion can be shown on a motion diagram as follows:

For constant acceleration, average acceleration over any time interval is equal to the instantaneous acceleration at any point along the x-axis. This means that,

…..(9)

Let the time interval start at time t₁ = 0 with an initial velocity, v_0x. Then at time t₂ = t, the velocity of the particle is given by v_x.

……(10)

Rearranging equation 10, the final velocity of the particle is given by;

……(11)

The final velocity of the particle is equal to the sum of initial velocity, v_0x and the product of x-acceleration with time interval, t. In other words, the change in x-velocity, v_x – v_0x is equal to a_xt.

This means that the change in x-velocity, v_x – v_0x is equal to the area under the curve between the time interval on an a_x-t graph.

Similarly, an equation can be derived for the position, x of the particle when acceleration, a_x is constant.

As discussed in equation 3, the average velocity of a particle over a time interval, Δt is given by,

Let the particle at t₁ = 0 be at position x₁ = x₀. After some time, at t₂ = t, the particle is located at x₂ = x. The average x-velocity is given by;

…..(12)

Note: Equation 12 is true regardless of whether the acceleration is constant or not.

The average velocity, v_av(x) is also equal to the average of the initial and final velocity,

…..(13)

Note: This expression is true only because acceleration is constant and velocity is increasing with a steady rate.

Proof

Consider a particle with initial velocity, v₀ at time, t₀ = 0. Given that the acceleration, a, is constant, an expression for the final velocity, v can be derived using equation 11;

v = v₀ + at…..(14)

where t = 1, 2, 3, 4, 5,………., n.

The average velocity for Δt = n-0 = n, can be calculated as follows;

The sum of n-numbers is given by,

Therefore, the average velocity, v_av is equal to the average of initial velocity, v₀ and final velocity, v₀ +an.

Equating equations 12 and 13,

which gives,

…..(14)

Equation 14 is applicable to use when it is known that acceleration is constant but the value is not given.

Using equation 11, v_x = v_0x + a_xt;

……(15)

……(16)

If in equation 16, the acceleration is zero, x = x₀ + v_0xt, which means that the displacement of the particle when velocity is constant is given by the product of velocity and time (v_0xt).

This means that when acceleration is not zero, the position of a particle, x is equal to the sum of:

• x₀: The initial location along the x-axis
• v_0xt: The displacement of the particle if acceleration was zero and velocity was constant
• (1/2)a_xt²: The displacement of the particle due to change in velocity caused by a non-zero acceleration

Comparing equation 15 to Figure 29, we can see that for constant acceleration, the area under the straight line on a v_x-t graph is equal to the change in position or displacement of the particle (x-x₀).

Note: The area under the a_x-t curve between the given time interval is always equal to the change in velocity even if acceleration is not constant. However, in that case, v – v_0x would not necessarily equal a_xt. Similarly, the area under the v_x-t curve between the given time interval always gives the change in position of the particle even if acceleration is not constant. However, equation 16 would no longer apply.

Since equation 16 is quadratic because the degree (highest power of the variable, t) is equal to two, the x-t graph for constant acceleration will always be a parabola.

To understand why the graph is a parabola for constant acceleration, consider the x-t graph for a particle with zero x-acceleration which means that it is moving with a constant velocity.

Finally, an equation of motion for constant acceleration that does not include time can be derived from equations discussed earlier.

This can be done by substituting the value of t given by equation 11 into equation 16;

Plugging the value of t into equation 16:

Using the algebraic identity (a+b)(a-b) = a²-b²,

or

…..(17)

Equations 11, 14, 16 and 17 are defined as the equations of motion with constant acceleration. Every single problem that includes straight line motion with constant acceleration can be solved using these equations.

Equations of Motion with Constant Acceleration

1. 
   • v_x is the final velocity of the particle at t = t
   • v_0x is the initial velocity of the particle at t=0
   • a_x is the value of constant acceleration
   • t is the time taken by the particle’s motion starting at t=0
2. 
   • x is the final position of the particle at t=t
   • x₀ is the initial position of the particle at t=0
   • v_0x is the initial velocity of the particle at t=0
   • v_x is the final velocity of the particle at t=t
   • t is the time taken by the particle’s motion starting at t=0
3. 
   • x is the final position of the particle at t=t
   • x₀ is the initial position of the particle at t=0
   • v_0x is the initial velocity of the particle at t=0
   • t is the time taken by the particle’s motion starting at t=0
   • a_x is the value of constant acceleration
4. 
   • v_x is the final velocity of the particle at t=t
   • v_0x is the initial velocity of the particle at t=0
   • a_x is the value of constant acceleration
   • x is the final position of the particle at t=t
   • x₀ is the initial position of the particle at t=0

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Free Falling Bodies (Young et al., 2016)

The most popular example of straight line motion with constant acceleration is that of a body falling under the force of gravity, commonly described to be in free fall. As theorized by Galileo, all bodies fall with a constant acceleration regardless of how much they weigh. This statement is true as long as air resistance is negligible.

Note: Acceleration of a falling body is constant with following assumptions:

• Air resistance is negligible
• Distance that the body falls is comparatively much smaller than the radius of the Earth
• Effects of Earth’s rotation are negligible

This constant acceleration is referred to as acceleration due to gravity and is usually taken to be g = 9.80m/s².

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Motion when Acceleration is not Constant (Young et al., 2016)

Velocity and Acceleration by differentiation

If acceleration is not constant,

Velocity can be determined from the position function¹³, x(t), using:

….(18)

X-velocity is the derivative of position function with respect to time.

Similarly, x-acceleration can be determined from the x-velocity function, v_x(t) using:

….(19)

X-acceleration is the derivative of x-velocity with respect to time.

Velocity and Position by Integration

When only the acceleration function is known, velocity and position functions can be derived using calculus, specifically, by the method of integration.

Consider a function of acceleration which varies with time, given by, a_x(t). If the time interval from t₁ to t₂ is split into equal mini-intervals of Δt and a_av(x) is the average acceleration for this time interval, Δt (as shown in Figure 34), then according to the definition of average acceleration;

or

….(20)

The change in x-velocity of the particle over the time interval Δt is equal to the product of the average x-acceleration and the time interval Δt. This means that the change in x-velocity is also equal to the area of the rectangle (as shown in Figure 34) which approximately equals the area under the curve between the given time interval, Δt.

The total change in x-velocity of the particle between times t₁ and t₂ (ΔV_x) is then equal to the sum of changes in x-velocity over each time interval Δt, which is approximately equal to the total area under the curve on an a_x-t graph (see Figure 34). In other words;

…..(21)

As Δt becomes really small and number of Δt’s become very large, the area of the rectangles approaches the area under the graph (See below). In other words, the average x-acceleration from any time t to t+Δt approaches the instantaneous x-acceleration, a_x(t) at time t.

In the limit that Δt becomes negligible and the number of Δt’s become infinite, the area under the curve is equal to the integral of instantaneous x-acceleration from t₁ to t₂. In other words,

The change in x-velocity of the particle is equal to the time integral of x-acceleration;

….(22)

If the initial velocity of the particle at t₁ = 0 is v_0x and the final velocity at t₂=t is v_x ;

….(23)

Similarly, the same procedure can be applied to a v_x-t graph such that change in the position (or displacement) of the particle is equal to the time integral of the x-velocity;

….(24)

The change in the position of the particle can then be written as;

….(25)

where x₀ is the initial position of the particle at t₁ = 0 and x is the final position of the particle at t₂ = t.

To summarize, the instantaneous velocity of the particle, v_x can be determined using equation 23 when acceleration is given as a function of time, a_x(t), and the initial velocity of the particle, v_0x is known. Once the velocity of the particle has been deduced, it can be used to determine the position, x, of the particle using equation 25 given that the initial position of the particle, x₀, is known.

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Common Errors in Understanding (Young et al., 2016)

Error #1:

Speed and Velocity are two different quantities. Speed is a scalar quantity which is given by distance travelled divided by the time taken. On the other hand, velocity is a vector quantity and is given by the particle’s displacement divided by the time interval.

The magnitude of average velocity is not the same as average speed.

However, magnitude of instantaneous velocity is the same as instantaneous speed. At a certain instant, instantaneous speed tells you how fast a particle is moving while instantaneous velocity tells you how fast and in what direction the particle is moving.

Back to Instantaneous Velocity

Error #2:

Whether the particle is speeding up or slowing down cannot be determined from the algebraic sign of instantaneous x-acceleration. Both instantaneous x-velocity and instantaneous x-acceleration need to be considered to find out about the particle’s speed. If v_x and a_x are in the same direction, particle is speeding up and if v_x and a_x are in the opposite direction, particle is slowing down.

Back to Instantaneous Acceleration

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1. Young, H.D. et al. (2016) Sears and Zemansky’s university physics: With modern physics. 14th edn. Boston: Pearson. ↩︎
2. Δx is not to be confused as the product between Δ and x. Δ is a symbol commonly used to describe change in a given quantity, whereby the change is equal to final value – initial value (not initial – final). ↩︎
3. Refrain from saying that since the particle moves to the right, displacement vector is in the positive direction. One could chose to have the -x direction pointed towards the right, in which case displacement vector will be negative if it is pointing towards the right. ↩︎
4. Change in time will always be positive because you can only move forward in time. ↩︎
5. The orange curve should not be mistaken to be the path taken by the particle, which is a straight line (yellow vector on the left). The curve represents the change in the particle’s position with respect to time. ↩︎
6. A particle can have a positive location (+x-coordinate) and have negative average or instantaneous velocity. The direction of average or instantaneous velocity does not depend on the location of the particle but instead depends on the change in it’s location (whether the change is along positive or negative x-axis). ↩︎
7. If the point is above the origin, it is located on the positive x-axis and if the point is somewhere below the origin, it is located on the negative x-axis. ↩︎
8. Instantaneous x-acceleration is referred to as simply ‘acceleration’ and instantaneous x-velocity is referred to as simply ‘velocity’. ↩︎
9. Numerical value of acceleration is hard to determine from an x-t curve because curvature cannot be measured easily. ↩︎
10. For constant acceleration, average and instantaneous acceleration will be the same. ↩︎
11. Graph of x-acceleration with respect to time is called an a_x-t graph. ↩︎
12. Alternatively, slope = rise/run = a_xt/t = a_x which makes sense because the slope of the line should equal the x-acceleration of the particle. ↩︎
13. Position is given as a function of time. ↩︎