Impulse Equation (College Board AP® Physics 1: Algebra-Based)

Study Guide

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Katie M

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Physics

Impulse equation

The change in an object’s motion due to an external force acting on it for a time is described by impulse
Impulse is defined as the product of the average force exerted and the time interval during which it acts

$\vec{J} = {\vec{F}}_{a v g} ∆ t$

Where:
- $\vec{J}$ = impulse, measured in $N \cdot s$
- ${\vec{F}}_{a v g}$ = average force exerted, measured in $N$
- $∆ t$ = time interval over which the force acts, measured in $s$
Impulse is a vector quantity, so it has magnitude and direction
Impulse has the same direction as the net force exerted on the system
Therefore, when the average force exerted over a time interval is zero $({\vec{F}}_{a v g} = 0)$ , no impulse is exerted on the system

Applications of impulse

The concept of impulse is often used to prevent injury
For example, vehicles contain many safety features, such as crumple zones, seat belts, and airbags,
These features are designed to minimize the force $({\vec{F}}_{a v g})$ exerted on a passenger during a collision by
- increasing the collision time $(∆ t)$ , as the impulse is constant
- increasing the distance $(d)$ over which kinetic energy $(∆ K)$ is dissipated (according to the Work-Energy Theorem)

Car safety features, including seat belts, airbags, and crumple zones. The seat belt and airbag increase the time over which the passenger experiences the impact force while the crumple zone absorbs energy which also reduces the impact force. — **The seat belt, airbag and crumple zones help reduce the risk of injury on a passenger by increasing the collision time**

Worked Example

How does a crash mat protect a gymnast from injury as they land on the ground during a gymnastics routine?

A It reduces the kinetic energy loss of the gymnast

B It reduces the momentum change of the gymnast

C It shortens the stopping time of the gymnast and increases the force applied during the landing

D It lengthens the stopping time of the gymnast and reduces the force applied during the landing

The correct answer is D

Answer:

Step 1: Analyze the scenario

During a landing, a force is exerted on the gymnast by the ground
The risk of injury can be reduced by minimizing the force exerted on the gymnast
When the gymnast lands on a crash mat, as opposed to the hard ground, the material compresses and lengthens the stopping time
This compression absorbs some of the kinetic energy and reduces the force exerted on the gymnast

Step 2: Eliminate the incorrect options

The change in kinetic energy is equivalent to the work done in deforming the mat, and this reduces the force by extending the distance over which energy is dissipated (since $∆ K = W = F d$ )
However, the kinetic energy loss of the gymnast depends on their speed before and after landing and is not reduced by the use of a crash mat
- This eliminates option A
The change in momentum of the gymnast also depends on their speed before and after landing and is not reduced by the use of a crash mat
- This eliminates option B

Step 3: Deduce the correct option

The crash mat increases the time taken for the gymnast to come to a stop
This decreases the acceleration (since $\vec{a} = \frac{∆ \vec{v}}{∆ t}$ )
This reduces the force exerted on the gymnast (since $\vec{J} = \vec{F} ∆ t$ and impulse is constant)
- Therefore, option D is correct

Examiner Tip

Avoid making the common mistake in which students consider impulse and force as equivalent. You must realize that impulse describes the length of time a force is exerted on an object.

We can see this just by looking at the unit of impulse, newton seconds or $N \cdot s$ . This is why $\vec{J} = \vec{F} ∆ t$ .

Furthermore, if we look at the base unit of impulse, we can see it is actually equivalent to momentum, or more specifically (as we will discover in the next study guide), change in momentum

Force: $1 N = 1 kg \cdot m / s^{2}$

Impulse: $1 N \cdot s = 1 kg \cdot m / s^{2} \times s = 1 kg \cdot m / s$

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