Introduction to Acceleration
Acceleration is an expression of the 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 is a vector quantity. Sometimes the magnitude of the acceleration vector is called acceleration , and is symbolized by lowercase italic letters such as a . Technically, however, the vector expression should be used; it is symbolized by lowercase boldface letters such as a .
In the example of a car driving along a highway, suppose that the speed is constant at 25 m/s (Fig. 7-7). 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 straightaway 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. 7.7 . Acceleration vectors x, y , and z for a car at three points ( X , Y , and Z ) along a road. Note that y is the zero vector because there is no acceleration at point Y .
Acceleration vectors can be illustrated graphically as arrows. In Fig. 7-7, three acceleration vectors are shown approximately for a car traveling along a 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 . Acceleration only takes place where the car is following a bend in the road. At point Y , the road is essentially straight, so the acceleration is zero. This 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 might seem esoteric at first. What is a second squared? Think of it in terms of a concrete example. Suppose that you have a car that can go from 0 to 60 miles per hour (0.00 to 60.0 mi/h) in 5 seconds (5.00 s). A speed of 60.0 mi/h is roughly equivalent to 26.8 m/s. 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 = (26.8 m/s)/(5 s) = 5.36 m/s 2
Of course, the instantaneous acceleration will not be constant in a real-life test of a car’s get-up-and-go power. However, the average acceleration magnitude will still be 5.36 m/s 2 —a speed increase of 5 meters per second with each passing second—assuming that the vehicle’s speed goes from 0.00 to 60.0 mi/h in 5.00 s.
Instantaneous acceleration magnitude can be measured in terms of the force it exerts on a known mass. This can be determined according to the amount of distortion in an elastic object such as a spring. The “force meter” shown in Fig. 7-4 can be adapted to make an “acceleration meter,” technically known as an accelerometer , for measuring acceleration magnitude. A fixed, known mass is placed in the device, and the deflection scale is calibrated in a laboratory environment. 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 will produce a force on the mass directly against the spring. A functional diagram of the basic arrangement is shown in Fig. 7-8.
A common spring scale can be used to measure acceleration indirectly. 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 intimately related, as we shall soon see.
Suppose that 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, that the magnitude of the displacement from the starting point is q . Then this formula applies:
q = a avg t 2 /2
In this example, suppose that 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
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