# Average acceleration: how exercises are calculated and solved

The ** mean acceleration** for

_{m}is the magnitude that describes the change in a particle’s velocity over time. It’s important because it shows the variations the movement experiences.

To express this magnitude in mathematical terms, it is necessary to consider two velocities and two instants of time, respectively designated as v _{1} and v _{2} , t _{1} and t _{2} .

Combining the values according to the given definition, the following expression will be used:

In the international SE system, the units for one _{m} will be m / s ^{2} , although other units involving length per unit of time squared will do.

For example, there is km/h which reads “kilometer per hour and per second”. Note that the time unit appears twice. Thinking of a cell phone that travels in a straight line means that, with every passing second, the cell phone increases its speed by 1 km/h. Or decreases by 1 km / h for every second that passes.

__Acceleration, speed and speed__

__Acceleration, speed and speed__

Although it is associated with acceleration with an increase in velocity, the truth is that, looking carefully at the definition, it turns out that any change in velocity implies the existence of an acceleration.

And velocity doesn’t always necessarily change in magnitude. It may happen that the cell phone only varies in direction and keeps the speed constant. There is still a responsible acceleration of this change.

An example of this is a car that turns at a constant speed of 60 km/h. The vehicle is subject to acceleration, responsible for changing the direction of speed so that the car follows the curve. The driver applies it using the steering wheel.

This acceleration is directed towards the center of the curved path, so that the car does not leave it. It is called **radial** or **normal** acceleration . If radial acceleration were suddenly canceled, the car could no longer continue the curve and continue in a straight line.

A car moving along a curve is an example of two-dimensional motion, whereas when it’s moving in a straight line, its motion is one-dimensional. In this case, the only effect of acceleration is to change the car’s speed.

This acceleration is called **tangential** acceleration . It is not exclusive to one-dimensional movement. The car that takes the curve at 60 km/h can simultaneously accelerate to 70 km/h while taking it. In this case, the driver needs to use the steering wheel and the accelerator pedal.

If we consider a one-dimensional motion, mean acceleration has a geometric interpretation similar to mean velocity, as a slope of the secant line that cuts the curve at points P and Q on the velocity versus time graph.

__How Average Acceleration Is Calculated__

__How Average Acceleration Is Calculated__

Let’s look at some examples to calculate the average acceleration in various situations:

I) At any given time, a mobile traveling in a straight line has a speed of +25 km / h and 120 seconds later it has another -10 km / h. What was the average acceleration?

**Response**

As the motion is one-dimensional, vector notation can be dispensed with; in this case:

v _{o} = +25 km / h = +6.94 m / s

v _{f} = -10 km / h = – 2.78 m / s

Δt = 120 s

Whenever you do a mixed magnitude exercise like this, where there are hours and seconds, you need to pass all the values into the same units.

Being a one-dimensional motion, vector notation was dispensed with.

II) A cyclist travels east at a rate of 2.6 m/s and 5 minutes later heads south at 1.8 m/s. Find your average acceleration.

**Response**

Motion is __not__ one-dimensional; therefore, use is made of vector notation. The unit vectors **i** and **j** indicate the directions with the following sign convention, facilitating the calculation:

- North: +
**j** - South: –
**j** - East: +
**i** - West: –
**i**

**v **_{2} = – 1.8 **j** m / s

**v **_{1} = + 2.6 **i** m / s

Δt = 5 minutes = 300 seconds

__One-dimensional motion acceleration signals__

__One-dimensional motion acceleration signals__

As always with average or average magnitudes, the information provided is global. They don’t offer details of what happened to the cell at any given time, but what they contribute is still valuable in describing the movement.

Through the speed and acceleration signals, it is possible to know if a cell phone that moves in a straight line accelerates or brakes. In both situations, acceleration is present as the velocity is changing.

Here are some interesting considerations about the signs of these two magnitudes:

- Average speed and acceleration, both of the same signal, means that, viewed globally, the cell phone is getting faster.
- Velocity and acceleration with different signals is a moving signal has greatly decreased.

It is often thought that whenever there is negative acceleration, the cell phone is stalling. This is true if the cell phone speed is positive. But if it’s negative, in fact the speed is increasing.

As always, when motion is studied, special cases are considered. For example, what happens when the average acceleration is zero? Does this mean that the cell phone has always kept its speed constant?

The answer is no. The cell phone could have varied its speed in the considered range, but the initial and final speed were the same. At the moment, the details of what happened in the interval are unknown, as the average acceleration does not provide more information.

What if the average acceleration *a _{m}* equals the acceleration

*a*at any point in the time interval? This is a very interesting situation called Uniformly Varied Rectilinear Movement or MRUV by its acronym.

This means the speed changes evenly over time. Therefore, the acceleration is constant. In nature there is this movement, with which everyone is familiar: the free fall.

__Free fall: a motion with constant acceleration__

__Free fall: a motion with constant acceleration__

It is known that the Earth attracts objects towards its center and that, when releasing some to a certain height, it experiences the acceleration of gravity, whose value is approximately constant and equal to 9.8 m / s ^{2} near the surface.

If air resistance does not interfere, the movement is vertical and is known as free fall. When the acceleration is constant and choosing t = 0, the mean acceleration equation is transformed into:

v _{f} = v + em = gt (v = 0)

Where a = g = 9.8 m / s ^{2}

__Exercise solved__

__Exercise solved__

An object falls from a sufficient height. Find speed after 1.25 seconds.

**Response**

v _{o} = 0, since the object was dropped, then:

v _{f} = gt = 9.8 x 1.25 m / s = 12.25 m / s, directed vertically towards the ground. (The vertical direction has been reduced to positive).

As the object approaches the ground, its speed increases by 9.8 m / s per elapsed second. The object’s mass is not involved. Two different objects, dropped from the same height at the same time, develop the same speed at which they fall.