Machinery's Handbook, 31st Edition
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Velocity and Acceleration VELOCITY, ACCELERATION, WORK, AND ENERGY Velocity and Acceleration
Motion is a progressive change of position of a body. Velocity is the rate of motion, that is, the rate of change of a body’s position. It is expressed in units of distance over time, such as feet per second, miles per hour, and kilometers per second. Uniform motion indicates that the velocity of a body is the same at every moment during which the motion takes place. When velocity is variable and constantly increasing, the rate at which it changes is called ac- celeration. Acceleration is the rate at which the velocity of a body changes in a unit of time, as the change of feet or meters per second in one second. When the motion is decreasing instead of increasing, it is called retarded motion, and the rate at which the motion is re- tarded is frequently called deceleration . If the acceleration is uniform, the motion is called uniformly accelerated motion. An example of such motion is that of falling bodies. Newton’s Laws of Motion.— The first clear statement of the fundamental relations exist ing between force and motion was made in the seventeenth century by Sir Isaac Newton, the English mathematician and physicist. It was put in the form of three laws, which are given as originally stated by Newton: 1) Every body continues in its state of rest, or uniform motion in a straight line, except in so far as it may be compelled by force to change that state. 2) Change of motion is proportional to the force applied and takes place in the direction in which that force acts. 3) To every action there is always an equal reaction; or, the mutual actions of two bodies are always equal and oppositely directed. Motion with Constant Velocity.— In the formulas that follow, S = distance moved; V = velocity; t = time of motion, θ = angle of rotation, and ω = angular velocity; the usual units for these quantities in the US customary system are, respectively, feet, feet per sec- ond, seconds, radians, and radians per second. The usual metric units are meters, meters per second, seconds, radians, and radians per second. Any consistent set of units may be employed. Constant Linear Velocity: S Vt = V S t ⁄ = t S V ⁄ = Constant Angular Velocity: θ ω t = ω θ t ⁄ = t θ ⁄ ω = Relation between Angular Motion and Linear Motion: The relation between the angular velocity of a rotating body and the linear velocity of a point at a distance r from the center of rotation is: V r ft/s ft radians/s # ~ = ^ ^ ^ h h h V r m/s m radians/s # ~ = ^ ^ ^ h h h Similarly, the distance moved by the point during rotation through angle θ is: S r ft ft radians # i = ^ ^ ^ h h h S r m m radians # i = ^ ^ ^ h h h Linear Motion with Constant Acceleration.— The relations between distance, velocity, and time for linear motion with constant or uniform acceleration are given by the formulas in the accompanying Table 1. In these formulas, the acceleration is assumed to be in the same direction as the initial velocity; hence, if the acceleration in a particular problem should happen to be in a direction opposite that of the initial velocity, then a should be replaced by − a . Thus, for example, the formula V f = V o + at becomes V f = V o − at when a and V o are opposite in direction. Example: A car is moving at 100 km/h when the brakes are suddenly locked and the car begins to skid. If it takes 2 seconds to slow the car to 50 km/h, at what rate is it being decelerated, how long is it before the car comes to a halt, and how far will it have traveled? Solution: The initial velocity V o of the car is 100 km/h or 27.78 m/s and the acceleration a due to braking is opposite in direction to V o , since the car is slowed to 50 km/h or 13.89 m/s.
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