13.5: Motion in 3-Space

University:
The Pennsylvania State University
Course:
Math 231 | Calculus Severable Variables
Academic year:

2024
Views:

197

Pages:

1
Author:

Alexis H.

Motion in 3-Space r (t + h) ¡ r (t) h!0 h v (t) = r0 (t) = lim As we have seen, v(t) points in the direction of motion (if it is nonzero), and its magnitude is the objects speed. The acceleration vector is the second derivative r”(t), which we shall denote a(t) If an object's acceleration is given, we can solve for v(t) and r(t) by integrating twice: v (t) = Z a (t) dt r (t) = Z v (t) dt Arbitrary constants arise in each of these antiderivatives. To determine specific functions v(t) and r(t), initial conditions need to be provided Near the surface of the earth, gravity imparts an acceleration of approximately 9.8 m/s^2 in the downward direction. This means that if we have a projective moving near the surface of the earth that has no additional means of acquiring acceleration, we know that its acceleration vector a(t) is determined by gravity In general, acceleration is the rate of change of velocity with respect to time. In linear motion, acceleration acceleration can be is zero if the speed is constant. By contrast, in two or three dimensions, the acceleration can be nonzero even when the objects speed is constant Understanding the Acceleration Vector Acceleration is the rate of change of velocity, and a velocity vector provides information about the direction of motion and the speed (via its magnitude). Thus acceleration can involve change in either the direction of the magnitude of velocity a = a TT + a NN aT = v 0 (t) aN = k (t) v (t) 2 The coefficient aT is called the tangential component and aN the normal component of acceleration Theorem 1: Tangential and normal a¢v aT = a ¢ T = kvk and a TT = ³a ¢ v ´ a¢v v aN = a ¢ N = q 2 kak ¡ jaTj a NN = a ¡ a TT = a ¡ ³a ¢ v ´ v¢v 2 V