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VF.4 Motion in Space: Velocity and Acceleration

In this section we show how the ideas of tangent and normal vectors and curvature can be used in physics to study the motion of an object, including its velocity and acceleration, along a space curve.

Suppose that a particle moves through space so that its position vector at time t is r(t )The velocity is defined in exactly the same way that the tangent vector is defined. It is called the velocity vector v (t) at time t:

Thus, the velocity vector is also the tangent vector and points in the direction of the tangent line. Also, the magnitude of the velocity vector gives the speed of the object moving along the space curve:

The acceleration of the particle is defined as the derivative of the velocity:

If the force that acts on a particle is known, then the acceleration an be found from Newton's Second Law of Motion. The vector version of this law states that if, at any time t, a force F(t) acts on an object of mass m producing an acceleration a(t), then:

Tangential and Normal Components of Acceleration

When we study motion of a particle, it is often useful to resolve the acceleration into two components, one in the direction of the tangent and the other in the direction of the normal. The general equation for this is:

Writing a and a for the tangential and normal components of acceleration, we have

So,

Some of the vectors outlined are shown below:

The above equations are good, however, one prefers that the equations be shown in terms of r and r'. We can express the components of acceleration in terms of r and r' as show below:

and

 

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