Acceleration
Acceleration is an expression of the rate of change in the velocity of an object. This can occur as a change in speed, a change in direction, or both. Acceleration can be defined in one dimension (along a straight line), in two dimensions (within a flat plane), or in three dimensions (in space), just as can velocity. Acceleration sometimes takes place in the same direction as an object’s velocity vector, but this is not necessarily the case.
Acceleration is a Vector
Acceleration, like velocity, is a vector quantity. Sometimes the magnitude of the acceleration vector is called “acceleration,” and is usually symbolized by the lowercase italic letter a. But technically, the vector expression should be used; it is normally symbolized by the lowercase bold letter a.
In our previous example of a car driving along a highway, suppose the speed is constant at 25 m/s. The velocity changes when the car goes around curves, and also if the car crests a hilltop or bottoms-out in a ravine or valley (although these can’t be shown in this two-dimensional drawing). If the car is going along a straight path, and its speed is increasing, then the acceleration vector points in the same direction that the car is traveling. If the car puts on the brakes, still moving along a straight path, then the acceleration vector points exactly opposite the direction of the car’s motion.
Fig. 15-7. Acceleration vectors x, y, and z for a car at three points (X, Y, and Z) along a road. The magnitude of y is 0 because there is no acceleration at point Y.
Acceleration vectors can be graphically illustrated as arrows. Figure 15-7 illustrates acceleration vectors for a car traveling along a level, but curving, road at a constant speed of 25 m/s. Three points are shown, called X, Y, and Z. The corresponding acceleration vectors are x, y, and z. Because the speed is constant and the road is level, acceleration only takes place where the car encounters a bend in the road. At point Y, the road is essentially straight, so the acceleration is zero (y = 0 ). The zero vector is shown as a point at the origin of a vector graph.
How Acceleration is Determined
Acceleration magnitude is expressed in meters per second per second, also called meters per second squared (m/s ^{2} ). This seems esoteric at first. What does s ^{2} mean? Is it a “square second”? What in the world is that? Forget about trying to imagine it in all its abstract perfection. Instead, think of it in terms of a concrete example. Suppose you have a car that can go from a standstill to a speed of 26.8 m/s in 5 seconds. Suppose that the acceleration rate is constant from the moment you first hit the gas pedal until you have attained a speed of 26.8 m/s on a level straightaway. Then you can calculate the acceleration magnitude:
a = (26.8m/s)/(5s) = 5.36 m/s ^{2}
The expression s ^{2} translates, in this context, to “second, every second.” The speed in the above example increases by 5.36 meters per second, every second.
Fig. 15-8. An accelerometer. This measures the magnitude only, and must be properly oriented to provide an accurate reading.
Acceleration magnitude can be measured in terms of force against mass. This force, in turn, can be determined according to the amount of distortion in a spring. The force meter shown in Fig. 15-4 can be adapted to make an acceleration meter, more technically known as an accelerometer, for measuring acceleration magnitude.
Here’s how a spring type accelerometer works. A functional diagram is shown in Fig. 15-8. Before the accelerometer can be used, it is calibrated in a lab. For the accelerometer to work, the direction of the acceleration vector must be in line with the spring axis, and the acceleration vector must point outward from the fixed anchor toward the mass. This produces a force on the mass. The force is a vector that points directly against the spring, exactly opposite the acceleration vector.
A common weight scale can be used to indirectly measure acceleration. When you stand on the scale, you compress a spring or balance a set of masses on a lever. This measures the downward force that the mass of your body exerts as a result of a phenomenon called the acceleration of gravity . The effect of gravitation on a mass is the same as that of an upward acceleration of approximately 9.8 m/s ^{2} . Force, mass, and acceleration are interrelated as follows:
F = m a
That is, force is the product of mass and acceleration. This formula is so important that it’s worth remembering, even if you aren’t a scientist. It quantifies and explains a lot of things in the real world, such as why it takes a fully loaded semi truck so much longer to get up to highway speed than the same truck when it’s empty, or why, if you drive around a slippery curve too fast, you risk sliding off the road.
Suppose an object starts from a dead stop and accelerates at an average magnitude of a _{avg} in a straight line for a period of time t . Suppose after this length of time, the distance from the starting point is d . Then this formula applies:
d = a _{avg} t ^{2} /2
In the above example, suppose the acceleration magnitude is constant; call it a. Let the instantaneous speed be called v _{inst} at time t . Then the instantaneous speed is related to the acceleration magnitude as follows:
v _{inst} = at
What is Acceleration? Practice Problems
Fig. 15-9. Illustration for Practice 1-5.
Practice 1
Suppose two objects, denoted by curves A and B in Fig. 15-9, accelerate along straight-line paths. What is the instantaneous acceleration a _{inst} at t = 4 seconds for object A?
Solution 1
The acceleration depicted by curve A is constant, because the speed increases at a constant rate with time. (That’s why the graph is a straight line.) The number of meters per second squared does not change throughout the time span shown. In 10 seconds, the object accelerates from 0 m/s to 10 m/s; that’s a rate of speed increase of one meter per second per second (1 m/s ^{2} ). Therefore, the acceleration at t = 4 seconds is a _{inst} = 1 m/s ^{2} .
Practice 2
What is the average acceleration a _{avg} of the object denoted by curve A in Fig. 15-9, during the time span from t = 2 seconds to t = 8 seconds?
Solution 2
Because the curve is a straight line, the acceleration is constant; we already know it is 1 m/s ^{2} . Therefore, a _{avg} = 1m/s ^{2} between any two points in time shown in the graph, including the two points corresponding to t = 2 seconds and t = 8 seconds.
Practice 3
Examine curve B in Fig. 15-9. What can be said about the instantaneous acceleration of the object whose motion is described by this curve?
Solution 3
The object starts out accelerating slowly, and as time passes, its instantaneous rate of acceleration increases.
Practice 4
Use visual approximation in the graph of Fig. 15-9. At what time t is the instantaneous acceleration a _{inst} of the object described by curve B equal to 1 m/s ^{2} ?
Solution 4
Use a straight-edge to visualize a line tangent to curve B whose slope is the same as that of curve A. Then locate the point on curve B where the line touches curve B. Finally, draw a line straight down, parallel to the speed ( v ) axis, until it intersects the time ( t ) axis. Read the value off the t axis. Here, it appears to be about t = 6.3 seconds.
Practice 5
Use visual approximation in the graph of Fig. 15-9. Consider the object whose motion is described by curve B. At the point in time t where the instantaneous acceleration a _{inst} is 1 m/s, what is the instantaneous speed, v _{inst} , of the object?
Solution 5
Locate the same point that you found in Problem 15-12, corresponding to the tangent point of curve B and the line parallel to curve A. Draw a horizontal line to the left, until it intersects the speed (v) axis. Read the value off the v axis. In this example, it looks like it’s about v _{inst} = 3.0 m/s.
Find practice problems and solutions for these concepts at the Movement Practice Test.