U3AOS1 Topic 10: Impulse

Introduction

Impulse is a fundamental concept in physics that describes the change in momentum of an object when a force is applied over a specific period. It plays a crucial role in understanding how forces influence the motion of objects and is essential in fields ranging from sports to engineering. Here’s a closer look at impulse, including its definition, formula, and practical applications.

Definition of Impulse

Impulse is defined as the product of the force applied to an object and the time duration over which the force acts. It effectively quantifies the impact of a force on an object's momentum. Mathematically, impulse $J$ can be expressed as:

$J = F \Delta t$

where $F$ is the force applied, and $\Delta t$ is the time interval during which the force acts.

Relationship to Momentum

Impulse is directly related to momentum, which is the product of an object's mass $m$ and its velocity $v$. The change in momentum ($\Delta p$) of an object is equal to the impulse applied to it. This relationship is expressed by the impulse-momentum theorem:

$J = \Delta p$

where $\Delta p = m \Delta v$ (change in momentum). Hence, impulse can also be written as:

$J = m \Delta v$

Calculation of Impulse

1. Constant Force:

   • If the force applied is constant over time, impulse is simply the product of this constant force and the time duration. For example, if a force of $10\ \text{N}$ is applied for $2\ \text{s}$, the impulse is: $J = F \Delta t = 10 \times 2 = 20\ \text{N}\cdot\text{s}$

2. Variable Force:

   • If the force varies with time, impulse can be calculated by integrating the force over time. For a force $F(t)$ that changes with time, the impulse is given by: $J = \int_{t_1}^{t_2} F(t) \, dt$

Practical Examples

1. Sports:

   • In sports, such as baseball or golf, players apply a force over a short duration to change the momentum of the ball. For instance, when a baseball bat hits a ball, the impulse imparted by the bat changes the ball’s momentum, determining its speed and direction.

2. Automobile Safety:

   • In vehicle crashes, crumple zones are designed to extend the time over which the collision occurs, thereby reducing the force experienced by passengers. This increase in collision time reduces the impulse imparted to the occupants, minimizing injury.

3. Rocket Propulsion:

   • Rockets work on the principle of impulse. The expulsion of gas at high speed generates a large impulse, which results in a reaction force that propels the rocket forward.

Impulse and the Change of Momentum

Impulse directly affects the change in an object's momentum. For instance, if a soccer ball is kicked with an impulse of $50\ \text{N}\cdot\text{s}$ and its mass is $0.5\ \text{kg}$, the change in velocity of the ball can be calculated by:

$\Delta v = \frac{J}{m} = \frac{50}{0.5} = 100\ \text{m/s}$

Thus, impulse provides a measure of how much the momentum of an object changes due to a force.

Impulse is the change in momentum and follows the equation

\[ \displaystyle \Huge I = \Delta \rho = m \Delta v = m(v-u)\]

$I = \Delta\rho = m\Delta v = m(v-u)$

Note: Impulse is also a vector.

Therefore the direction matters!!

Impulse can also be calculated by:

\[ \displaystyle \Huge I=F \Delta t\]

$I = F\Delta t$