8.2 Impulse

Impulse and Its Applications

Impulse measures the effect of a force acting over a period of time. It connects force directly to changes in motion, which makes it essential for analyzing collisions, designing safety features, and understanding propulsion. The core idea is that the same change in motion can come from a large force over a short time or a small force over a long time.

Impulse and the Force-Time Relationship

Impulse is the product of the average net force on an object and the time interval during which that force acts:

$J = F_{avg} \cdot \Delta t$

• $F_{avg}$ is the average net force (in newtons)
• $\Delta t$ is the time interval (in seconds)

Impulse is a vector quantity, pointing in the same direction as the average net force. Its SI unit is the newton-second (N·s), which is equivalent to kg·m/s.

Because impulse depends on both force and time, the same impulse can result from very different situations. A large force over a short time produces the same impulse as a small force over a long time. This tradeoff is the basis for most real-world applications of impulse.

Real-World Applications of Impulse

Car safety features work by increasing the time over which a collision happens, which reduces the average force on passengers. Crumple zones at the front of a car deform gradually during a crash, stretching the impact from a few milliseconds to tens of milliseconds. Airbags do something similar for your head and torso. Seatbelts also spread the stopping force across a longer time interval and over a larger area of your body.

Sports equipment uses the same principle. Padded boxing gloves and football helmets increase the contact time during an impact, lowering the peak force on the body. On the offensive side, a tennis racket's flexible strings keep the ball in contact with the racket longer, increasing the impulse delivered to the ball and sending it off at higher speed.

Rocket propulsion generates a large impulse by expelling exhaust gas at high velocity over an extended time. The longer a rocket engine fires and the faster it expels gas, the greater the total impulse, which is what allows the rocket to overcome gravity and accelerate.

Force-Time Graphs and Impulse Analysis

On a force-time graph, the area under the curve equals the impulse. This works regardless of the shape of the curve.

• For a constant force (rectangular graph), impulse is simply $F \cdot \Delta t$.
• For a linearly changing force (triangular graph), impulse is $\frac{1}{2} \cdot F_{peak} \cdot \Delta t$.
• For more complex curves, you'd integrate the force function over the time interval, or estimate the area by counting grid squares or using geometric approximations.

You can also find the average force from a graph by rearranging:

$F_{avg} = \frac{J}{\Delta t}$

This gives you the constant force that would produce the same impulse over the same time interval.

Problem-Solving with Impulse

The impulse-momentum theorem states that impulse equals the change in an object's momentum:

$J = \Delta p = m \cdot \Delta v$

• $m$ is the object's mass
• $\Delta v$ is the change in velocity (final minus initial)

Combining this with the force-time definition gives you the key equation for most problems:

$F_{avg} \cdot \Delta t = m \cdot \Delta v$

This equation has four quantities. If you know any three, you can solve for the fourth. Here's a typical approach:

1. Identify the object and the time interval during which the force acts.
2. Determine the object's velocity before and after the force acts to find $\Delta v$.
3. Plug known values into $F_{avg} \cdot \Delta t = m \cdot \Delta v$ and solve for the unknown.

For example, if a 0.15 kg baseball goes from rest to 40 m/s and the bat is in contact for 0.002 s, the average force is:

$F_{avg} = \frac{0.15 \times 40}{0.002} = 3000 \text{ N}$

When two objects collide in an isolated system (no external net force), the impulse on one object is equal in magnitude and opposite in direction to the impulse on the other. This is a direct consequence of Newton's third law and is why total momentum is conserved in such systems.

Collisions and Impulse

Collisions are interactions where objects exert impulses on each other, changing both objects' momenta.

• Elastic collisions: Both momentum and total kinetic energy are conserved. The objects bounce apart. A good approximation is two billiard balls colliding.
• Inelastic collisions: Momentum is conserved, but some kinetic energy is converted into other forms (heat, sound, deformation). If the objects stick together after the collision, it's called a perfectly inelastic collision.

The coefficient of restitution ($e$) quantifies how "bouncy" a collision is. It's defined as the ratio of relative speed after the collision to relative speed before. A value of $e = 1$ means perfectly elastic (no kinetic energy lost), and $e = 0$ means perfectly inelastic (the objects stick together). Most real collisions fall somewhere in between.