3.6 Impulse | Translatory Motion | Physics 11

Introduction

In physics, impulse quantifies the overall effect of a force acting over a period of time. It is particularly useful for analyzing situations involving large forces that act for very short durations, such as a bat hitting a ball or a hammer striking a nail. Impulse is directly related to the change in an object's momentum.

Definition and Formula

Impulse ( $I$ ) is defined as the product of the average force ( $F_{a v g}$ ) acting on an object and the time interval ( $Δ t$ ) over which that force acts.

Formula: $I = F_{a v g} \times Δ t$

Vector Quantity: Since force is a vector, impulse is also a vector quantity, and its direction is the same as the direction of the average force.
Units: The SI unit for impulse is the Newton-second (N·s). In terms of base units, it is equivalent to $kg \cdot m \cdot s^{- 1}$ .
Dimensions: The dimensions of impulse are $[M L T^{- 1}]$ , which are the same as the dimensions of momentum.

Derived Units→ Dimensions→

The Impulse-Momentum Theorem

The most important relationship involving impulse is the Impulse-Momentum Theorem. This theorem states that the impulse applied to an object is equal to the change in that object's momentum ( $Δ p$ ).

Derivation:

We start with Newton's Second Law of Motion: $F = m a$

We know that acceleration ( $a$ ) is the rate of change of velocity: $a = \frac{Δ v}{Δ t} = \frac{v _{f} - v _{i}}{Δ t}$

Substituting this into Newton's Second Law: $F = m (\frac{v _{f} - v _{i}}{Δ t})$

Multiplying both sides by the time interval $Δ t$ : $F Δ t = m (v_{f} - v_{i})$

The left side is the definition of impulse, and the right side is the change in momentum ( $m v_{f} - m v_{i} = p_{f} - p_{i} = Δ p$ ).

Therefore, we arrive at the Impulse-Momentum Theorem: $I = Δ p$

This confirms that the units of impulse (N·s) are equivalent to the units of momentum (kg·m/s).

Graphical Representation

In a Force vs. Time graph, the area under the curve represents the impulse.

For a constant force, this area is a simple rectangle ( $F \times Δ t$ ).
For a variable force, the area under the curve represents the total impulse. This graphical method is very useful for visualizing the total effect of a force that changes over time, which is common in real-world collisions.

Applications and Significance

The concept of impulse helps explain how the same change in momentum can be achieved in different ways: $F_{a v g} \cdot Δ t = Δ p$

Scenario	Strategy	Effect
Karate chop breaking a board	Large $F$ , short $Δ t$	Large impulse delivered quickly
Airbag in a car	Small $F$ , long $Δ t$	Same $Δ p$ , reduced injury force
Bending knees on landing	Increases $Δ t$	Reduces average force on joints
Baseball glove padding	Increases $Δ t$	Reduces sting of catch

Large Force, Short Time: In a collision like a karate chop breaking a board, a very large force is applied for a very short time to produce the required change in momentum.
Small Force, Long Time: To minimize the force of impact, the time of the collision can be extended. This is the principle behind airbags in cars, bending your knees when landing from a jump, and the padding in a baseball glove. In each case, $Δ t$ is increased, which reduces $F_{a v g}$ for the same change in momentum.

Difference Between Impulse and Momentum

Property	Momentum ( $p$ )	Impulse ( $I$ )
Definition	$p = m v$	$I = F_{a v g} Δ t$
Nature	State of a moving object at an instant	Change produced by a force over time
Role	The "effect"	The "cause" of change in momentum
Units	$kg \cdot m \cdot s^{- 1}$	N·s (equivalent)

Introduction

Definition and Formula

Impulse ( $I$ ) is defined as the product of the average force ( $F_{a v g}$ ) acting on an object and the time interval ( $Δ t$ ) over which that force acts.

Formula: $I = F_{a v g} \times Δ t$

Vector Quantity: Since force is a vector, impulse is also a vector quantity, and its direction is the same as the direction of the average force.
Units: The SI unit for impulse is the Newton-second (N·s). In terms of base units, it is equivalent to $kg \cdot m \cdot s^{- 1}$ .
Dimensions: The dimensions of impulse are $[M L T^{- 1}]$ , which are the same as the dimensions of momentum.

Derived Units→ Dimensions→

The Impulse-Momentum Theorem

Derivation:

We start with Newton's Second Law of Motion: $F = m a$

We know that acceleration ( $a$ ) is the rate of change of velocity: $a = \frac{Δ v}{Δ t} = \frac{v _{f} - v _{i}}{Δ t}$

Substituting this into Newton's Second Law: $F = m (\frac{v _{f} - v _{i}}{Δ t})$

Multiplying both sides by the time interval $Δ t$ : $F Δ t = m (v_{f} - v_{i})$

The left side is the definition of impulse, and the right side is the change in momentum ( $m v_{f} - m v_{i} = p_{f} - p_{i} = Δ p$ ).

Therefore, we arrive at the Impulse-Momentum Theorem: $I = Δ p$

This confirms that the units of impulse (N·s) are equivalent to the units of momentum (kg·m/s).

Graphical Representation

In a Force vs. Time graph, the area under the curve represents the impulse.

For a constant force, this area is a simple rectangle ( $F \times Δ t$ ).
For a variable force, the area under the curve represents the total impulse. This graphical method is very useful for visualizing the total effect of a force that changes over time, which is common in real-world collisions.

Applications and Significance

The concept of impulse helps explain how the same change in momentum can be achieved in different ways: $F_{a v g} \cdot Δ t = Δ p$

Scenario	Strategy	Effect
Karate chop breaking a board	Large $F$ , short $Δ t$	Large impulse delivered quickly
Airbag in a car	Small $F$ , long $Δ t$	Same $Δ p$ , reduced injury force
Bending knees on landing	Increases $Δ t$	Reduces average force on joints
Baseball glove padding	Increases $Δ t$	Reduces sting of catch

Large Force, Short Time: In a collision like a karate chop breaking a board, a very large force is applied for a very short time to produce the required change in momentum.
Small Force, Long Time: To minimize the force of impact, the time of the collision can be extended. This is the principle behind airbags in cars, bending your knees when landing from a jump, and the padding in a baseball glove. In each case, $Δ t$ is increased, which reduces $F_{a v g}$ for the same change in momentum.

Difference Between Impulse and Momentum

Property	Momentum ( $p$ )	Impulse ( $I$ )
Definition	$p = m v$	$I = F_{a v g} Δ t$
Nature	State of a moving object at an instant	Change produced by a force over time
Role	The "effect"	The "cause" of change in momentum
Units	$kg \cdot m \cdot s^{- 1}$	N·s (equivalent)