NEGATIVE ACCELERATION: FORMULAS, CALCULATION AND SOLVED EXERCISES - PHYSICAL - 2025

The negative acceleration arises when the change or variation in speed over a period of time has a negative sign. Observe the dog in the image having fun on the beach. The sand slows his movement, which means that there is an acceleration opposite to the speed he is carrying.

This acceleration can be considered negative, as opposed to velocity, which is stated as positive. Although a negative acceleration does not always cause a reduction in speed.

The dog has fun braking in the sand. A braking acceleration can be considered negative acceleration. Source: Pixabay.

In a one-dimensional movement, the direction of advance is generally taken as positive, that is, the direction of speed. This is what has been considered previously: in the dog in the figure, the positive direction is the one that goes from the tail to the head.

Before sinking its paws into the sand, the dog came with a certain speed v forward, that is, positive. Then the sand slows down until it stops, that is, zero final speed.

Suppose that all of this occurred in a time period Δt. The acceleration during this time will be calculated like this:

In the above equation v> 0, Δt> 0 then a <0, that is negative acceleration (a <0). Since the direction of velocity at the start was taken as positive, then negative acceleration means that the acceleration is pointing away from the velocity. Therefore it is concluded that:

Therefore we can establish that:

• When velocity and acceleration have the same sign, regardless of which sign, the speed increases. In such a case the velocity becomes more positive or more negative, depending on the case.
• When velocity and acceleration have opposite signs, speed decreases.

Formulas for calculating acceleration

Regardless of the sign it has, the average acceleration a _m between instants t and t 'is calculated by the following formula:

The average acceleration provides global information about how the speed has changed in the time interval considered. For its part, the instantaneous acceleration offers the detail of how the speed changes at each moment. So for a given instant t, the acceleration is calculated by the following formula:

-Example 1

At the initial instant t = 0.2 s an object has a velocity of 3 m / s. Later, at the instant t '= 0.4 s, it has a speed of 1 m / s. Calculate the mean acceleration between times t and t 'and interpret the result.

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-Example 2

At the initial instant t = 0.6 s an object has velocity -1 m / s. Subsequently at the instant t '= 0.8 s it has a speed of -3 m / s. Calculate the mean acceleration between times t and t '. Interpret the result.

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In conclusion, at the end of the time interval the velocity became even more negative (-3m / s).

Does this mean that the mobile slows down its movement? No. The minus sign in the speed only means that it is going backwards and faster, since going at -3 m / s is going faster than at -1m / s, the speed that was at the beginning.

The speed, which is the modulus of the velocity, has increased despite having negative acceleration. I mean, this object sped up. Therefore we conclude:

-Example 3-The vertical throw up

Consider the following example: an object has instantaneous velocity given by the following expression, with all units in the International System:

Find the velocity and acceleration for times 0s, 0.5s, and 1.0s. In each case, indicate whether the object is accelerating or decelerating.

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The velocity at each of the indicated instants is found by substituting t directly into the equation. Acceleration is found by deriving the given expression as a function of time and then evaluating the result at each of the given times.

The results are the following:

Acceleration is constant and negative for all motion. It is now possible to describe what has happened to the mobile as it moves.

At time t = 0 s the mobile was slowing down. This follows immediately since the velocity is positive and the acceleration is negative.

At the instant t = 0.5 s the mobile stopped, at least momentarily it was at rest. It is not impossible for a mobile to stop even when it is accelerated. The most palpable example is in the vertical upward throw.

Graduates pitch vertically toward their caps. Source: Pexels.

When a mobile is projected vertically upwards, it manages to reach a maximum height. If the positive direction is chosen in that sense, which is what is almost always done, during the time it takes to reach that maximum point, the mobile will have positive velocity.

But gravity has been around the entire time. And it is always directed vertically downwards, no matter whether the object is going up or down. Naturally she succeeds in making the mobile slow down gradually until it stops for an instant.

The mobile immediately reverses its speed and heads back to the ground. In this case the velocity is negative because it also points towards the ground. Therefore gravity causes the speed to increase more and more.

The value of the acceleration of gravity has been estimated at 9.8 m / s ², which for calculation purposes is rounded to 10 m / s ². The object in the example may well have been thrown upward with an initial velocity of 5 m / s.

Finally at t = 1.0 s, the velocity of the mobile is negative. If it is a vertical upward launch, in the absence of friction, it means that it is passing through the starting point again, but this time it is going downwards, instead of upwards.

In conclusion, a negative acceleration does not necessarily mean that the mobile is slowing down. On the contrary, the mobile could go faster and faster. It is a matter of paying attention to whether the signs of speed and acceleration are the same or not.

References

1. Walker, J. 2010. Physics. Fourth edition. Addison Wesley. 26-30.