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Average and Instantaneous Acceleration

Many time the following questions are asked about a car, how fast can the car go from zero to 60 Mi/H? or how long will take the car to stop, if the car is moving at the speed of 100 Km/H? Even when the answer to these questions may involve a calculation of time, these calculations are made using the concept of acceleration (or deceleration). Just as how fast the change in the displacement was studied throughout the study of velocity, the change in velocity can also be studied through a new quantity called acceleration. The definition and properties of acceleration are studied below.

The truck above is in motion with constant acceleration until the velocity reaches 150 miles/hour. The time represented in the clock is real time but the velocity is not to scale. In addition, when the truck reaches the right most side of the screen, the truck reappears on the left most side of the screen. 
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Definition of Average Acceleration,

The average acceleration measures how fast the velocity is changing per unit of time,

The average acceleration is a vector pointing in the direction of the change in velocity. Thus, the average acceleration does not point in the direction of the initial velocity nor the final velocity; rather, it points in the direction of the change in velocity. In the above equation, both the initial and final velocities are the instantaneous velocity measured at two different times.

In the case of the truck going over the small hill, the direction of the acceleration is the same as the direction of the green vector. The direction of this vector is different than the direction of either of the velocities. Also, notice that the magnitude of the initial velocity and the final velocity are the same; however, there is  acceleration just from the change in direction.

There is not a quantity equivalent to the acceleration defined in terms of the speed rather than the velocity. Remember that the instantaneous speed is just the magnitude of the instantaneous velocity.

Units

The units of acceleration are derived from the units of velocity and the unit of time. In the MKS system, the units of acceleration are the . The corresponding unit in the British engineering system is the ft/s2 (feet/second square). Unit such as kilometer per hour square are almost never used because the impracticability of observing regular every day motions  that present changes in velocity for extended periods of times. Think about a car coming from rest after a traffic light, how long is the car accelerating? 5 s, 10 s, maybe 50 s, never a long time. The tachometer in the car does not measure acceleration, it actually measures the revolutions per minute of the engine. Even when there is a relation between the revolutions of the car engine and the acceleration of the car, this relations is not true at different speed and with different gears. Most regular cars do not have accelerometers.

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Instantaneous Acceleration

Because of cars not having accelerometers, it is harder to visualize the concept of instantaneous acceleration. Nevertheless, the instantaneous acceleration is defined following the same approach used when defining the instantaneous velocity. That it is, consider the average acceleration defined above for the case in which the change in time gets smaller and smaller to the limiting case when Dt → 0. In this case, we have the definition of instantaneous acceleration,

Graphically, the instantaneous acceleration is the slope of the line tangent to the curve of the velocity versus the time. In the figure at the left, the black line represent a line tangent to the curve (red) at a single point. The slope of this line can be used for calculating the instantaneous speed. Notice that the slope of this line can be calculated using two convenient points on the line. In this case the slope may be calculated using the points selected in the figure (completely arbitrary points),

 

In order to describe motion, the acceleration is the last quantity necessary to be studied. In other words, it is not necessary to study the change in the instantaneous acceleration. Why? The answer to this question is that the physical quantity that produce the motion is related to the acceleration. This association is presented in Newton's Second Law.

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by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com