# Acceleration

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**Acceleration**

When a particle’s velocity changes, the particle is said to undergo acceleration (or to accelerate). For motion along an axis, the average acceleration over a time interval is

where the particle has velocity 1′, at time t, and then velocity 1’2 at time t2.The instantaneous acceleration (or simply acceleration) is the derivative of the velocity with respect to time

In words, the acceleration of a particle at any instant is the rate at which its velocity is changing at that instant. Graphically, the acceleration at any point is the slope of the curve of 1′(1) at that point.

We can combine Eq. 2-8 with Eq. 2-4 to write

In words, the acceleration of a panicle at any instant is the second derivative of its position x(t) with respect to time.

A common unit of acceleration is the meter per second per second: m/(s . s) or m/s2. You will see other units in the problems, but they will each be in the form of length/(time ‘ time) or length/time”. Acceleration has both magnitude and direction (it is yet another vector quantity). Its algebraic sign represents its direction on an axis just as for displacement and velocity; that is, acceleration with a positive value is in the positive direction of an axis, and acceleration with a negative value is in

the negative direction.

Figure 2-6c is a plot of the acceleration of the elevator cab discussed in Sample Problem 2-2. Compare this a(t) curve with the vet) curve-each point on the a(t) curve shows the derivative (slope) of the I'(t) curve at the corresponding time. When v is constant (at either 0 or 4 m/s), the derivative is zero and so also is the acceleration, When the cab first begins to move, the vet) curve has a positive derivative (the slope is positive), which means that a (t) is positive. When the cab slows to a stop,

the derivative and slope of the vet) curve are negative; that is, (t) is negative.

Next compare the slopes of the vet) curve during the two acceleration periods, The slope associated with the cab’s slowing down (commonly called “deceleration) is steeper, because the cab stops in half the time it took to get up to speed. The steeper slope means that the magnitude of the deceleration is larger than that of the acceleration, as indicated in.

The sensations you would feel while riding in the cab of Fig. 2-6 are indicated by the sketched figures, When the cab first accelerates, you feel as though you are pressed downward; when later the cab is braked to a stop, you seem to be stretched upward, In between, you feel nothing special, Your body reacts to. accelerations (it is an accelerometer) but not to velocities (it is not a speedometer). When you are in a car traveling at 90 km/h or an airplane traveling at 900 km/h, you have no bodily

awareness of the motion. However, if the car or plane quickly changes velocity, you may become keenly aware of the change, perhaps even frightened by it. Part of the thrill of an amusement park ride is due to the quick changes of velocity that you undergo (you pay for the accelerations, not for the speed). A more extreme example is shown in the photographs of Fig. 2-7, which were taken while a rocket sled was rapidly accelerated along a track and then rapidly braked to a stop.

Large accelerations are sometimes expressed in terms of g units, with

(As we shall discuss in Section 2-8, g is the magnitude of the acceleration of a faIling object near Earth’s surface.) On a roller coaster, you may experience brief accelerations up to 3g, which is (3)(9.8 m/s2) or about 29 m/s2, more than enough to justify the cost of the ride.