How to Calculate the Rate of Change of Momentum
Understanding the rate of change of momentum is fundamental to classical mechanics and physics. It provides a quantitative measure of the force required to change an object's motion over a specific period. This concept is the cornerstone of Newton's Second Law of Motion.
What is Rate of Change of Momentum?
Momentum (symbolized as p) is the quantity of motion of a moving body, measured as a product of its mass and velocity. The Rate of Change of Momentum describes how quickly this momentum changes over time.
According to Newton's Second Law, the rate of change of momentum of a body is directly proportional to the applied force. In mathematical terms, calculating the rate of change of momentum is equivalent to calculating the average Force ($F$) acting on the object.
The Formula
The calculation involves the object's mass, its change in velocity, and the time duration of that change. The formula is derived as follows:
F = (mvf – mvi) / t
F = m(vf – vi) / t
Where:
- m = Mass of the object (kg)
- vf = Final velocity (m/s)
- vi = Initial velocity (m/s)
- t = Time interval (seconds)
- F = Rate of change of momentum (Newtons)
Example Calculation
Let's look at a realistic example of a soccer player kicking a ball.
- Mass of ball: 0.45 kg
- Initial Velocity: 0 m/s (ball is stationary)
- Final Velocity: 30 m/s (after the kick)
- Contact Time: 0.05 seconds
Step 1: Calculate Initial Momentum
$0.45 \text{ kg} \times 0 \text{ m/s} = 0 \text{ kg}\cdot\text{m/s}$
Step 2: Calculate Final Momentum
$0.45 \text{ kg} \times 30 \text{ m/s} = 13.5 \text{ kg}\cdot\text{m/s}$
Step 3: Calculate Change in Momentum
$13.5 – 0 = 13.5 \text{ kg}\cdot\text{m/s}$
Step 4: Calculate Rate of Change
$13.5 / 0.05 \text{ s} = 270 \text{ Newtons}$
The average force exerted by the player's foot on the ball is 270 Newtons.
Why is this Important?
Calculating the rate of change of momentum is critical in engineering and safety design. For example, in car crash safety, airbags and crumple zones are designed to extend the Time ($t$) of the collision. By increasing the time it takes for a passenger's velocity to reach zero, the rate of change of momentum (and thus the Force) is reduced, minimizing injury.