Now, what happens when the speed is constant i.e., what happens when `|vec v|=c`, where `c` is a constant? Well, in this case, the square of the speed will be `|vec v|^2=c^2`, so the square of the speed is also constant. But the magnitude of the vectors does not change i.e., the speed is constant. Note that the velocity is not constant, because the direction of the velocity vectors in Figure 10 is always changing. Speed is the magnitude of velocity, so the speed of the particle traveling on the helix in Figure 10 is There is a reason why the acceleration vectors are orthogonal to the velocity vectors in the current example, the reason being that the position vector `vec r(t)=langle cos(t), sin(t), t//(8 pi) rangle` traces what is know as constant speed curve.īecause `vec r(t)=langle cos(t), sin(t), t//(8 pi) rangle`, we saw that the velocity is given by This was not the case in the elliptical example of Figure 7. In Figure 10, note that all of the acceleration vectors are orthogonal (perpendicular) to the velocity vectors. Thus, the depiction of the acceleration vector in Figure 4 seems reasonable.Īdding acceleration vectors to the path. Therefore, the acceleration of the particle must point in that same direction. This cannot happen unless an external force is pushing the particle in that direction. Newton's first law states that "an object in motion tends to stay in motion with the same speed and direction unless acted upon by an external force." Note how the path of the particle bends downward. Furthermore, because the mass `m` is a positive scalar quantity, the vectors `vec F` and `vec a` must point in the same direction. This means that the force vector `vec F` is a scalar multiple of the acceleration vector `vec a`, which means that the vectors `vec F` and `vec a` must be parallel. To take the second derivative, we simply take the second derivative of each component of the position vector (reasons for this are discussed in your text). However, in the case of motion in the plane, the position is a vector valued function of time. The velocity vector points in the direction of motion and is tangnet to the path at each instant of time.Īcceleration: As in the case of motion along the line, the second derivative of the position with respect to time yields the acceleration. Therefore, the velocity vector should be tangent to the path of the particle at that instant in time, as shown in Figure 3. Consequently, the velocity vector should point in the direction of motion at that instant. The result should be a vector that determines the instantaneous velocity at a particular moment in time. To take the derivative, we simply take the derivative of each component of the position vector (reasons for this are discussed in your text). Velocity: As in the case of motion along a line, the first derivative of the position with respect to time yields the velocity. The vector `vec r(t)` changes with time, both in magnitude and direction, and as it changes, the tip of the position vector traces out the path of the particle shown in Figure 2. That is, the vector `vec r` gives us the position of the particle as a function of time. Position: The position of the particle is determined by the position vector `vec r(t) = langle x(t), y(t) rangle`. The tip of the position vector traces the path of the particle as a function of time.
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