Impulse and Momentum: Physics I Study Notes

Impulse and Momentum

Momentum

Momentum is a fundamental property of matter that describes the quantity of motion an object possesses. It is defined for an object of mass m and velocity v as:

• Definition: $\vec{p} = m \vec{v}$

• Vector Nature: Since velocity is a vector, momentum is also a vector and points in the same direction as velocity.

• Units: The SI unit for momentum is kg·m/s.

• Alternate Names: Momentum is also called linear momentum or translational momentum.

Change of Momentum

The change of momentum (also called impulse) is the difference between the final and initial momentum of an object. Since momentum is a vector, vector subtraction must be used.

• Formula: $\Delta \vec{p} = \vec{p}_{\text{final}} - \vec{p}_{\text{initial}} = m (\vec{v}_{\text{final}} - \vec{v}_{\text{initial}})$

• Example: If an object reverses direction, the change in momentum can be significant even if the speed is similar.

• Key Point: Always pay attention to the direction (sign) when calculating changes in momentum.

Connecting Force and Momentum

Newton's Second Law can be expressed in terms of momentum. By differentiating momentum with respect to time, we connect force and momentum:

• $\frac{d\vec{p}}{dt} = \frac{d}{dt}(m\vec{v}) = m \frac{d\vec{v}}{dt} = m\vec{a}$

• From Newton's Second Law: $m\vec{a} = \sum \vec{F} = \vec{F}_{\text{net}}$

• Momentum Form of Newton's Second Law: $\sum \vec{F} = \vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}$

Impulse-Momentum Theorem

The impulse delivered to an object is the integral of the net force over the time interval during which the force acts. The impulse-momentum theorem relates impulse to the change in momentum:

• Impulse: $\vec{J} \equiv \int \vec{F}_{\text{net}} dt$

• Impulse-Momentum Theorem: $\vec{J} = \int \vec{F}_{\text{net}} dt = \Delta \vec{p}$

• Units: Impulse has units of Newton-seconds (N·s) or kg·m/s.

Worked Example: Calculating Final Velocity Using Impulse

Problem: A 0.5 kg object is subjected to a varying force in the +x direction from $t = 0$ to $t = 7$ s, starting from rest. What is its final velocity?

• Given: Force vs. time graph with points at (1,4), (2,2), (3,5), (4,1), (6,3), (7,0).

• Solution Steps:

  1. Calculate the area under the force-time graph to find the impulse (total area = 18 N·s).

  2. Impulse equals change in momentum: $J = \Delta p = p_{\text{final}} - p_{\text{initial}}$

  3. Since the object starts from rest, $p_{\text{initial}} = 0$, so $p_{\text{final}} = 18$ kg·m/s.

  4. Final velocity: $v_{\text{final}} = \frac{p_{\text{final}}}{m} = \frac{18}{0.5} = 36$ m/s

Key Take-Away Concepts

1. Momentum of an object: $\vec{p} = m\vec{v}$

2. Newton's Second Law (momentum form): $\sum \vec{F} = \vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}$

3. Impulse: $\vec{J} = \int \vec{F} dt$

4. Impulse-Momentum Theorem: $\vec{J} = \int \vec{F} dt = \Delta \vec{p}$

Practice Problem

Problem: A 0.375 kg rubber ball traveling horizontally to the right at 10 m/s hits a wall and bounces back with a velocity of 6 m/s to the left. What is the total impulse exerted by the wall on the ball?

• Solution Outline:

  1. Initial momentum: $p_i = m v_i = 0.375 \times 10 = 3.75$ kg·m/s (right)

  2. Final momentum: $p_f = 0.375 \times (-6) = -2.25$ kg·m/s (left)

  3. Impulse: $J = p_f - p_i = -2.25 - 3.75 = -6.0$ N·s

• Correct Answer: A) -6.0 N·s

Laboratory Activity: Verifying the Impulse-Momentum Theorem

Experimental Setup

• Apparatus: Cart on a track, force probe, motion detector, LoggerPro software.

• Calibration: Calibrate force probe at two points (0 and 0.49 N).

• Measurement: Measure velocity and force during the experiment.

Analysis

• Calculate the change in momentum of the cart using measured change in velocity.

• Use software to integrate the force vs. time graph to find impulse.

• Compare the total impulse (including friction) with the change in momentum.

Optional Material: Special Theory of Relativity

Einstein's Two Postulates

• First Postulate: The laws of physics are the same in all inertial frames.

• Second Postulate: The speed of light, c, is constant in all inertial frames.

Inertial Frames

• An inertial frame is a reference frame in which Newton's First Law holds (no acceleration or rotation).

• Laboratory setups are typically inertial frames; rotating or accelerating systems are not.

Einstein's Correction to Newton's Second Law

• At high velocities, Newton's Second Law must be modified. The relativistic momentum is:

$\vec{p} = \frac{m \vec{v}}{\sqrt{1 - \frac{v^2}{c^2}}}$

• As speed approaches the speed of light, the denominator approaches zero, so momentum increases without bound.

• At everyday speeds (much less than c), the correction is negligible.

Additional info: The relativistic form of momentum ensures consistency with the principle that no object with mass can reach or exceed the speed of light.