To find velocity in vector form, you calculate the derivative of the position vector with respect to time.
In physics and calculus, the motion of an object is often described using vector functions. The position of a particle at any given time t can be represented by a position vector, typically denoted as r(t). This vector points from the origin of a coordinate system to the particle's location.
The Position Vector r(t)
In three-dimensional space, the position vector r(t) is usually expressed in terms of its components along the x, y, and z axes:
r(t) = x(t)^i + y(t)^j + z(t)^k
Here, x(t), y(t), and z(t) are scalar functions of time that describe the particle's coordinates, and i, j, and k are the standard unit vectors along the x, y, and z axes, respectively.
The Definition of Velocity
Velocity is the rate at which the position of an object changes with respect to time. In the context of vector functions, this rate of change is found by taking the derivative of the position vector.
As defined: "Let r(t) be a differentiable vector valued function representing the position vector of a particle at time t. Then the velocity vector is the derivative of the position vector. v(t)=r′(t)=x′(t)^i+y′(t)^j+z′(t)^k."
This means that to find the velocity vector v(t), you differentiate each component of the position vector r(t) with respect to time t.
How to Calculate the Velocity Vector
Finding the velocity vector involves applying the rules of differentiation to each scalar component function of the position vector:
- Start with the position vector: Identify r(t) = x(t)^i + y(t)^j + z(t)^k.
- Differentiate each component: Calculate the derivative of x(t) with respect to t to get x′(t). Similarly, find y′(t) and z′(t).
- Form the velocity vector: Combine the derivatives to form the velocity vector v(t) using the formula:
v(t) = x′(t)^i + y′(t)^j + z′(t)^k
Each component x′(t), y′(t), and z′(t) represents the instantaneous rate of change of the particle's coordinate along the respective axis.
Example
Suppose the position of a particle at time t is given by the vector function:
r(t) = (t² + 1)^i + (3t)^j + (sin(t))^k
To find the velocity vector v(t), we differentiate each component with respect to t:
- x(t) = t² + 1
x′(t) = d/dt (t² + 1) = 2t - y(t) = 3t
y′(t) = d/dt (3t) = 3 - z(t) = sin(t)
z′(t) = d/dt (sin(t)) = cos(t)
So, the velocity vector is:
v(t) = (2t)^i + (3)^j + (cos(t))^k
This vector gives the velocity of the particle at any specific time t. For instance, at t = 1, the velocity vector is v(1) = (2(1))^i + 3^j + (cos(1))^k = 2^i + 3^j + cos(1)^k.
Summary Table
Here's a simple table illustrating the relationship:
| Concept | Vector Representation | How to Find It |
|---|---|---|
| Position | r(t) = x(t)^i + y(t)^j + z(t)^*k | Given or derived from initial conditions/acceleration |
| Velocity | v(t) = vₓ(t)^i + vᵧ(t)^j + v₂(t)^*k | v(t) = r′(t) = x′(t)^i + y′(t)^j + z′(t)^*k |
Understanding the relationship between the position vector and its derivative allows you to describe and analyze the motion of objects in space.
