Rate of Motion

Measuring The Rate Of Motion

Measuring the rate of motion involves quantifying how quickly an object changes its position. This is primarily done using the concepts of speed and velocity.

Speed With Direction

When we talk about "speed with direction," we are referring to the concept of velocity. Velocity is a vector quantity that describes both how fast an object is moving and in what direction it is moving.

• Velocity: It is defined as the rate of change of displacement with respect to time. Mathematically, for average velocity:

  $ \vec{v}_{\text{avg}} = \frac{\Delta \vec{x}}{\Delta t} = \frac{\text{displacement}}{\text{time interval}} $

  where $ \Delta \vec{x} $ is the displacement and $ \Delta t $ is the time interval over which the displacement occurred.
• Speed: Speed is the magnitude of velocity. It tells us how fast an object is moving, but not its direction. For average speed:

  $ \text{Average Speed} = \frac{\text{Total Path Length}}{\text{Time Interval}} $

  If an object moves in a straight line without changing direction, its speed is the magnitude of its velocity. However, if the direction changes, the average speed can be different from the magnitude of the average velocity.
• Instantaneous Velocity: This is the velocity of an object at a specific moment in time. It is found by taking the limit of the average velocity as the time interval approaches zero:

  $ \vec{v}(t) = \lim_{\Delta t \to 0} \frac{\Delta \vec{x}}{\Delta t} = \frac{d\vec{x}}{dt} $

  This is the instantaneous rate of change of position, and it includes both the instantaneous speed and the direction of motion at that moment.

Example: A car traveling at 60 km/h East has a velocity of 60 km/h East. Its speed is simply 60 km/h. If the car then turns and travels at 60 km/h North, its speed remains 60 km/h, but its velocity has changed because the direction has changed.

Rate Of Change Of Velocity

The rate at which an object's velocity changes is described by the concept of acceleration. Acceleration is also a vector quantity because velocity is a vector quantity.

• Acceleration: It is defined as the rate of change of velocity with respect to time. Mathematically, for average acceleration:

  $ \vec{a}_{\text{avg}} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\text{change in velocity}}{\text{time interval}} $

  where $ \Delta \vec{v} $ is the change in velocity ($ \vec{v}_f - \vec{v}_i $) and $ \Delta t $ is the time interval.
• Instantaneous Acceleration: This is the acceleration of an object at a specific moment in time. It is found by taking the limit of the average acceleration as the time interval approaches zero:

  $ \vec{a}(t) = \lim_{\Delta t \to 0} \frac{\Delta \vec{v}}{\Delta t} = \frac{d\vec{v}}{dt} $

What does acceleration mean?

• Change in Speed: If an object's speed increases or decreases, it is accelerating. For example, a car speeding up from 0 to 60 km/h experiences acceleration.
• Change in Direction: Even if an object's speed is constant, if its direction of motion changes, it is still accelerating. For instance, a car moving in a circle at a constant speed has a velocity that is constantly changing direction, meaning it has acceleration (centripetal acceleration) directed towards the center of the circle.

Relationship to Force: According to Newton's second law of motion, acceleration is directly proportional to the net force acting on the object and inversely proportional to its mass: $ \vec{F}_{\text{net}} = m\vec{a} $. Therefore, acceleration is the measure of how the applied forces are causing the object's velocity to change.

Example: A ball dropped from rest. Initially, its velocity is $ 0 $. After 1 second, its velocity might be $ 9.8 \, \text{m/s} $ downwards. After 2 seconds, it might be $ 19.6 \, \text{m/s} $ downwards. The velocity is changing due to gravity. The acceleration due to gravity is approximately $ 9.8 \, \text{m/s}^2 $ downwards, indicating the rate at which the downward velocity is increasing.