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Instantaneous Acceleration: Definition, Formula and more

In this article, you will learn what we mean by instantaneous acceleration, or more simply acceleration, when describing the motion of a particle.

We will see the definition and formula for instantaneous acceleration with an example that demonstrates how to use the formula in practice.

We will also cover other important things that you should know, like what the instantaneous acceleration is represented by on a velocity vs time graph.

Definition and formula for instantaneous acceleration

The acceleration a that a particle has at an instant t is equal to the value that the average acceleration, calculated for an interval of time Δt which includes the instant t, approaches as the interval of time Δt gets smaller and smaller, i.e., as Δt approaches 0.

We know that the average acceleration a for an interval of time Δt is expressed as:

aΔv
Δt

where Δv is the change in velocity that occurs during Δt.

The formula for the instantaneous acceleration a is almost the same, except that we need to indicate that we're interested in knowing what the ratio of Δv to Δt approaches as Δt approaches zero.

We can indicate that by using the limit notation.

So, the formula for the instantaneous acceleration is:

alim Δv
Δt0Δt

To demonstrate how to use this formula in practice, let's go through a simple example.

Let's consider a particle whose velocity (in meters per second) at an instant t (in seconds) is given by 2t2:

v = 2t2

So, at 1 s the velocity is 2 m/s, at 2 s the velocity is 8 m/s, at 3 s the velocity is 18 m/s, etc.

Let's say that we want to find the acceleration of the particle at the instant t = 3 s.

The first thing we need to do is to choose an interval of time Δt which includes the instant 3 s.

This interval of time Δt starts at some instant t1 and ends at some instant t2 such that

t13 st2

For simplicity, we will choose t1 = 3 s so that t1 is as close as possible to 3 s and Δt can be made smaller by choosing values of t2 that are closer to 3 s.

t1 = 3 s
t2 > 3 s

Let's begin by choosing t2 equal to 3.1 s.

t1 = 3 s
t2 = 3.1 s
Δt = t2t1 = 3.1 s3 s = 0.1 s

The average acceleration for Δt is equal to:

aΔv = v2v1
Δtt2t1

Let's find the velocity v1 at instant t1:

v1 = 2t12
v1 = 2 (3)2 m/s
v1 = 18 m/s

and the velocity v2 at instant t2:

v2 = 2t22
v2 = 2 (3.1)2 m/s
v2 = 19.22 m/s

Now, we can calculate the average acceleration:

av2v1
t2t1
a19.22 m/s18 m/s
3.1 s3 s
a1.22 m/s
0.1 s
a = 12.2 m/s2

So, when the interval of time Δt is between 3 s and 3.1 s, the average acceleration is 12.2 m/s2.

Let's see what happens when we choose a smaller interval of time Δt, with t2 equal to 3.01 s.

t1 = 3 s
t2 = 3.01 s
Δt = 3.01 s3 s = 0.01 s

We already know that the velocity v1 at instant t1 is 18 m/s.

The velocity v2 at instant t2 is:

v2 = 2t22
v2 = 2 (3.01)2 m/s
v2 = 18.1202 m/s

The average acceleration is:

av2v1
t2t1
a18.1202 m/s18 m/s
3.01 s3 s
a0.1202 m/s
0.01 s
a = 12.02 m/s2

So, when the interval of time Δt is between 3 s and 3.01 s, the average acceleration is 12.02 m/s2.

One last time, let's choose an even smaller Δt, with t2 equal to 3.001 s.

t1 = 3 s
t2 = 3.001 s
Δt = 0.001 s

The velocity v2 at instant t2 is:

v2 = 2t22
v2 = 2 (3.001)2 m/s
v2 = 18.012002 m/s

The average acceleration is:

a18.012002 m/s18 m/s
3.001 s3 s
a0.012002 m/s
0.001 s
a = 12.002 m/s2

So, as Δt is getting smaller and smaller, the average acceleration seems to be approaching 12 m/s2.

We can keep choosing a smaller and smaller Δt ad infinitum and get closer and closer to 12 m/s2.

However, we can show that the average acceleration approaches 12 m/s2, as Δt gets smaller and smaller, in a more rigorous way so that we can be sure that the acceleration at the instant 3 s is 12 m/s2.

First of all, we will be referring to the instant 3 s as t to indicate that what we're about to do, not only applies to the specific instant 3 s, but to any instant t.

We begin, as before, by considering an interval of time Δt which starts at some instant t1 and ends at some instant t2 such that

t1tt2

Once again, we will choose t1 = t so that t1 is as close as possible to t and Δt can be made smaller by choosing values of t2 that are closer to t.

t1 = t
t2 > t

Since Δt = t2 − t1 and t1 = t, we can write:

Δt = t2t
t2 = t + Δt

This means that the boundaries of the interval of time Δt are the instant t and the instant t + Δt.

The velocity vt at instant t is:

vt = 2t2

and the velocity vt+Δt at instant t + Δt is:

vt+Δt = 2 (t + Δt)2
vt+Δt = 2 (t2 + 2tΔt + Δt2)
vt+Δt = 2t2 + 4tΔt + 2Δt2

Now, we can calculate the average acceleration for Δt:

avt+Δtvt
Δt
a2t2 + 4tΔt + 2Δt22t2
Δt
a4tΔt + 2Δt2
Δt
a = 4t + 2Δt

Notice that this expression, 4t + 2Δt, explains why the average acceleration that we were manually computing before was 12.2 m/s2 when Δt was 0.1 s, 12.02 m/s2 when Δt was 0.01 s, and 12.002 m/s2 when Δt was 0.001 s:

(4 × 3 + 2 × 0.1) m/s2 = 12.2 m/s2
(4 × 3 + 2 × 0.01) m/s2 = 12.02 m/s2
(4 × 3 + 2 × 0.001) m/s2 = 12.002 m/s2

Now, remember that the instantaneous acceleration is equal to the value that the average acceleration approaches as the interval of time Δt approaches 0:

alim Δv
Δt0Δt
alim vt+Δtvt
Δt0Δt

In this case, we found that the average acceleration is given by 4t + 2Δt. So,

alim 4t + 2Δt
Δt0

As Δt approaches 0, the term 2Δt, within the expression 4t + 2Δt, approaches 0, so the expression approaches 4t.

Therefore, the instantaneous acceleration a is given by 4t:

a = 4t

So, the acceleration of the particle at any instant t is given by 4t.

Thus, at the instant t = 3 s, the acceleration is 4 × 3 m/s2, i.e., 12 m/s2.

Instantaneous acceleration as derivative

The limit

lim vt+Δtvt
Δt0Δt

is called the derivative of v with respect to t, which is written as

dv
dt

Therefore, we say that the instantaneous acceleration is the derivative of velocity with respect to time:

adv
dt

Furthermore, since velocity is the derivative of position with respect to time:

vdx
dt

We can write the instantaneous acceleration as:

adv = d dx = d2x
dtdtdtdt2

Thus, the instantaneous acceleration is the second derivative of position with respect to time:

ad2x
dt2

Acceleration vs time graphs

Often, to show how the acceleration of a particle changes over time, an acceleration vs time graph is used.

Here's what an acceleration vs time graph might look like for a moving particle:

Acceleration vs time graph; the acceleration at time 0 is 0, then becomes positive, and finally, at 9 seconds, it returns back to 0.a210864t(s)897654321O2)(m/s

In this case, at the instant t = 4 s, the particle has an acceleration a = 6 m/s2:

Acceleration vs time graph; at 4 seconds, the acceleration is 6 meters per second squared.a210864t(s)897654321O2)(m/s

Recall that in a previous example, we found that the instantaneous acceleration of a particle was given by 4t:

a = 4t

Let's show how the acceleration of this particle changes over time with an acceleration vs time graph.

The graph of the function 4t is a line so we only need to find two points to draw it.

To keep the calculations simple, we will choose the instant t1 = 0 s and the instant t2 = 1 s:

t1 = 0 s, a1 = (4 × 0) m/s2 = 0
t2 = 1 s, a2 = (4 × 1) m/s2 = 4 m/s2

Now that we found two points, we are ready to draw the acceleration vs time graph:

Acceleration vs time graph consisting of a line that passes through the points (0, 0) and (1, 4).a210864t(s)321O2)(m/s

Velocity vs time graphs and instantaneous acceleration

Instantaneous acceleration as the slope of a tangent line to the velocity vs time graph

Let's consider a velocity vs time graph for the motion of a particle:

Velocity vs time graph; the velocity at time 0 is 0, then becomes positive, and finally goes back to 0.vtO

Remember that a velocity vs time graph shows how the velocity of the particle changes over time.

As we saw before, when we want to find the acceleration of the particle at an instant t, we consider an interval of time Δt, which starts at t and ends at t + Δt:

Velocity vs time graph; at instant t, the velocity is v sub-t; at instant t + Δt, the velocity is v sub-t + Δt; Δv and Δt are indicated.vtOtΔt+tttΔt+ΔtΔvvv

The acceleration at the instant t is equal to whatever the average acceleration for Δt approaches as Δt approaches zero.

We can use this information to determine what the instantaneous acceleration is represented by on a velocity vs time graph.

First, let's draw a secant line that passes through the points t and t + Δt on the graph:

Velocity vs time graph; the secant line that passes through the points t and t + Δt on the graph.vtOttΔt+ΔttΔt+tΔvvv

The slope of the secant line is equal to the average acceleration for the interval of time Δt because Δv/Δt represents both the average acceleration and the slope of the secant line.

Thus, the acceleration at the instant t is equal to what the slope of the secant line approaches as Δt approaches zero.

As Δt gets smaller and smaller, the secant line gets closer and closer to the line tangent to the graph at the point t:

Velocity vs time graph; as Δt approaches 0, the secant line approaches the tangent line at the point t on the graph.ΔvvtOttΔt+ΔttΔt+tvv

So, as Δt approaches 0, the slope of the secant line approaches the slope of the line tangent to the graph at the point t.

Therefore, the acceleration at an instant t is equal to the slope of the line tangent to the velocity vs time graph at the point t.

Velocity vs time graph; the line tangent to the graph at the point t.vtOt

If we label the slope of the tangent line with mT, then we can write

adv = mT
dt

How to determine the sign of the instantaneous acceleration from a velocity vs time graph

Determining the sign of the instantaneous acceleration from a velocity vs time graph is quite simple.

To show how to do that, let's return to the same velocity vs time graph that we've seen before and let's consider an instant t at which we want to know the sign of the acceleration:

Velocity vs time graph; at instant t, the velocity is v sub-t.vtOttv

We already know that the acceleration at the instant t is equal to the slope of the line tangent to the graph at the point t.

Let's represent the tangent line and let's call θ the angle that it makes with the positive t-axis:

Velocity vs time graph; the angle θ that the tangent line at the point t makes with the positive t-axis.vtOttvθ

To determine whether the acceleration at the instant t is positive, negative, or zero, we just have to look at the sign of the angle θ.

When the angle θ is positive:

Velocity vs time graph; the angle θ, that the tangent line makes with the positive t-axis, is positive.vtOttvθ

The slope of the tangent line is positive, and therefore the instantaneous acceleration is positive.

When the angle θ is negative:

Velocity vs time graph; the angle θ, that the tangent line makes with the positive t-axis, is negative.vtOttvθ

The slope of the tangent line is negative, and therefore the instantaneous acceleration is negative.

Finally, when the angle θ is zero:

Velocity vs time graph; the tangent line is a horizontal line.vtOttv

The slope of the tangent line is zero, and therefore the instantaneous acceleration is zero.

Thus, the sign of the acceleration at an instant t is the same as that of the angle θ that the line, tangent to the velocity vs time graph at the point t, makes with the positive t-axis.

So, we can easily determine when the acceleration is positive, negative, and zero, just by looking at the angle θ at different points on a velocity vs time graph:

Velocity vs time graph; initially the acceleration is positive, then, for a moment it becomes zero, and finally, it becomes negative.vtOa0>a0<a0=

What is deceleration?

When acceleration causes velocity to decrease in magnitude, it is sometimes called deceleration.

So, when a particle has a positive velocity and a negative acceleration or a negative velocity and a positive acceleration, you may hear the acceleration of the particle be called deceleration.

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