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acceleration, the time rate at which the velocity of a point is changing. Because velocity, being a vector quantity, has both magnitude and direction, acceleration is also a vector quantity and must account for changes in both the magnitude and direction of a velocity. The velocity of a point is represented graphically by a vector (arrow) having a length to some scale, such as one inch equals 30 kilometres per hour, proportional to the magnitude of the velocity and pointing along the tangent to the path; e.g., on a circular path, it is perpendicular to the radius of the circle. On a straight path, the velocity can change in magnitude only; on a curved path, it may or may not change in magnitude, but it will always change in direction. This condition means that the acceleration of a point moving on a curved path can never be zero.
If the velocity of a point moving on a straight path is increasing (i.e., if the speed, which is the magnitude of a velocity, is in- creasing), the acceleration vector will point in the same direction as the velocity vector. If the velocity is decreasing, the acceleration vector will point in the opposite direction. The average acceleration during a time interval is equal to the total change in the velocity during the interval divided by the time interval.
The acceleration at any instant is equal to the velocity change in the interval following the instant in question divided by the time interval, as the time interval approaches zero. To carry out this process accurately, the differential calculus is necessary, but good approximations to instantaneous accelerations can be obtained by using small time intervals.
When a point moves on a curved path, the acceleration that results from the change in the direction of the velocity vector is the same whether the point is moving at a constant speed or not. It is always perpendicular to the velocity vector and pointing inward on the concave side of the path; its magnitude is given by the square of the velocity divided by the radius of curvature r of the path: v2/r. The change in the magnitude of v is represented by another vector collinear with v and in the same direction ii v is increasing and the opposite direction if v is decreasing. If velocity is measured in metres per second, acceleration will be in metres per second per second.
When speed is decreasing, it is said to be decelerating.
