Question

A body of mass \(m\) moves along a trajectory \(\vec{r}(t)\) in three-dimensional space with constant kinetic energy, What geometric relationship has to exist between the body's velocity vector, \(\vec{v}(t),\) and its acceleration vector, \(\vec{a}(t),\) in order to accomplish this

Short answer

Answer: The geometric relationship between the velocity vector and the acceleration vector in this case is that they should be orthogonal or perpendicular to each other, meaning the angle between them is 90 degrees. This ensures that the kinetic energy remains constant.

Step-by-step solution

Write the expression for kinetic energy

For a body of mass \(m\) moving with velocity \(\vec{v}(t)\), its kinetic energy, \(K\), can be written as: K = \frac{1}{2}m\|\vec{v}(t)\|^2.

Differentiate the kinetic energy expression

To determine when the kinetic energy is constant, we need to find when its derivative with respect to time is zero. We differentiate \(K\) with respect to \(t\): \frac{dK}{dt} = \frac{d}{dt}\left(\frac{1}{2}m\|\vec{v}(t)\|^2\right).

Use the chain rule in calculus

Applying the chain rule, we get: \frac{dK}{dt} = m\|\vec{v}(t)\|\frac{d\|\vec{v}(t)\|}{dt}. Now, we notice that the derivative of \(\|\vec{v}(t)\|\) with respect to \(t\) can be related to acceleration. Since the acceleration \(\vec{a}(t) = \frac{d\vec{v}(t)}{dt}\), we see that: \frac{d\|\vec{v}(t)\|}{dt} = \frac{\vec{v}(t) \cdot \vec{a}(t)}{\|\vec{v}(t)\|}, where \(\cdot\) denotes the dot product.

Substitute in the derivative expression

Substituting this result into the expression for \(\frac{dK}{dt}\), we get: \frac{dK}{dt} = m\vec{v}(t) \cdot \vec{a}(t).

Determine the geometric relationship

Recall that we want to find the relationship that will make the derivative of kinetic energy equal to zero. Therefore, for constant kinetic energy, we must have: \frac{dK}{dt} = m\vec{v}(t) \cdot \vec{a}(t) = 0. This means that the velocity vector \(\vec{v}(t)\) and the acceleration vector \(\vec{a}(t)\) should be orthogonal or perpendicular to each other, that is, their dot product should be zero: \vec{v}(t) \cdot \vec{a}(t) = 0. This geometric relationship indicates that the angle between the velocity and acceleration vectors is 90 degrees, making their dot product zero. To maintain constant kinetic energy along the trajectory, the velocity and acceleration vectors must always be perpendicular to each other.

Key concepts

Velocity and Acceleration

When discussing the motion of objects, two crucial components to understand are velocity and acceleration. Velocity is a vector that tells us how fast something is moving and in what direction. It is calculated by taking the change in position over time. Acceleration, on the other hand, measures the rate of change of velocity over time. Simply put, it shows how quickly the velocity is changing. For an object whose speed and direction are changing, acceleration is present. - Velocity gives us speed and direction. - Acceleration gives us the change in velocity over time. An interesting relationship occurs when an object's kinetic energy is constant. The velocity does not change in magnitude, but the direction can still change. This means there is acceleration, but the way it acts in relation to velocity is particularly special.

Orthogonal Vectors

In mathematics and physics, vectors are often described as orthogonal when they are perpendicular to each other. This means they form a 90-degree angle. The dot product of two orthogonal vectors is zero. This is a crucial concept when discussing the relationship between velocity and acceleration as seen in this exercise.When the velocity vector \(\vec{v}(t)\) and the acceleration vector \(\vec{a}(t)\) of a body are orthogonal, their dot product equals zero: \(\vec{v}(t) \cdot \vec{a}(t) = 0\). - Orthogonality implies a 90-degree angle between vectors.- Orthogonal vectors have a dot product of zero.In the context of constant kinetic energy, having these vectors orthogonal ensures the speed (or magnitude) of velocity doesn't change over time, just its direction does. This keeps the kinetic energy constant while allowing directional changes in motion.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed. Kinetic energy is the energy of motion and, for a body moving with a constant kinetic energy, this principle implies that no work is being done to increase or decrease this energy. When velocity and acceleration vectors are orthogonal: - The kinetic energy remains constant. - There is no change in the magnitude of velocity, only direction. For a body, the conservation of kinetic energy ensures a balanced state where it neither speeds up nor slows down, despite potential changes in direction. This highlights an intriguing aspect of motion where constant energy does not mean static movement but rather a balance where forces involved result in orthogonal velocity and acceleration vectors. The conservation of energy principle acts as a guiding star in many areas, emphasizing balance and continuity in physical systems.