Illustration 3.3: The Direction of Velocity and Acceleration Vectors
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A putted golf ball rolls across the green as shown in the animation (position is given in meters and time is given in seconds). The animation represents the top view of the motion of the ball. Restart.
What is the direction of the velocity of the golf ball at any instant of time? View the velocity vector to check your answer (notice that a black line tangent to the object's path is also drawn). The direction of the velocity vector is determined by a fairly simple rule: It is always tangent to the path and in the direction of motion. The "direction of motion" is basically the direction of the object's displacement during a very small time interval. Since the displacement divided by this very small time interval approaches the instantaneous velocity (see Illustration 2.3), the instantaneous velocity must point in the direction of motion. This also directly follows from the definition of instantaneous velocity as the derivative of the position vector with respect to time.
What about the acceleration vector? It points in the direction of the change in velocity during any small time interval. This is again a result of the definition of acceleration. However, it also follows an interesting rule.
The acceleration vector can be resolved into two components, a component tangent to the path (called the tangential acceleration) and a component perpendicular to the path (called the radial component). View the velocity and acceleration vectors. The radial component of the acceleration is related to the change in direction of the velocity vector and points along the radius of curvature. The tangential component of the acceleration is related to the change in the magnitude of the velocity vector. In other words, it is related to the change in the speed. If the object is slowing down, then the tangential component of the acceleration is opposite to the velocity. If the object is speeding up, the tangential component of the acceleration is in the same direction as the velocity.
Click here to view the velocity vector (blue), acceleration vector (orange), and acceleration components (yellow for the tangential component and red for the radial component).
Illustration authored by Aaron Titus with support by the National Science Foundation under Grant No. DUE-9952323 and placed in the public domain.