Conservation of Momentum Study Guide

Two-Particle Collisions and Newton's Third Law

• Physical Interactions during Collisions: When two objects (such as the balls described in Figure 9) collide, each exerts a brief force on the Other. According to Newton's Third Law of Motion, these forces are equal in magnitude and opposite in direction. This is represented by the formula $F_{\text{red on blue}} = -F_{\text{blue on red}}$.

• Impulse Comparison: Because the time interval ($\Delta t$) over which these forces act is identical for both objects, the impulses imparted are also equal in magnitude but opposite in direction: $(F\Delta t)_{\text{red on blue}} = -(F\Delta t)_{\text{blue on red}}$.

• Mass and Velocity Independence: These equalities (force and impulse) hold true regardless of differences in the objects' masses, sizes, or initial velocities. No detail of mass is required to establish that these interaction forces are equal and opposite.

Momentum in Closed and Isolated Systems

• Impulse-Momentum Theorem Application: The change in momentum is equal to the impulse. In a collision between Ball C and Ball D:

  • For ball C: $P_{Cf} - P_{Ci} = F_{\text{D on C}} \Delta t$

  • For ball D: $P_{Df} - P_{Di} = F_{\text{C on D}} \Delta t$

• Vector Sum of Momentum: Since the impulses are equal and opposite, the sum of momenta before the collision equals the sum after the collision: $P_{Ci} + P_{Di} = P_{Cf} + P_{Df}$.

• Conservation Principle: This implies that the momentum gained by one object (Ball D) is exactly equal to the momentum lost by the other (Ball C). Thus, the net change in system momentum is zero if the system is defined as the two objects.

• System Definitions:

  • Closed System: A system that does not gain or lose mass.

  • Isolated System: A closed system where the net external force is zero (the only forces involved are internal to the system).

  • Real World Note: No system on Earth is perfectly isolated due to constant interactions with surroundings, but these often are small enough to be disregarded in physics problems.

• Law of Conservation of Momentum: The momentum of any closed, isolated system does not change. This allows for calculations of states before and after interactions without needing internal details of the collision itself.

Collision Examples and Practice Data

• Example Problem 3 (Stick-Together Collision):

  • Scenario: A $1875\,kg$ car (Car C) moving at $+23\,m/s$ hits a $1025\,kg$ car (Car D) moving at $+17\,m/s$ in the same direction. They stick together.

  • Formula: $m_C v_{Ci} + m_D v_{Di} = (m_C + m_D)v_f$

  • Calculation: $\frac{(1875\,kg)(+23\,m/s) + (1025\,kg)(+17\,m/s)}{1875\,kg + 1025\,kg} = +21\,m/s$

  • Evaluation: The result is reasonable as it falls between the two initial speeds and is closer to the speed of the more massive car.

• Practice Problem 17: Two freight cars ($3.0 \times 10^4\,kg$ each); one moving at $2.2\,m/s$, one at rest. Resulting speed when sticking: $1.1\,m/s$.

• Practice Problem 18: A $0.105\,kg$ hockey puck at $24\,m/s$ caught by a $75\,kg$ goalie at rest. Resulting speed: $\approx 0.034\,m/s$.

• Practice Problem 19: A $35.0\,g$ bullet ($0.035\,kg$) at $475\,m/s$ strikes a stationary $2.5\,kg$ steel ball. The bullet bounces back at $5.0\,m/s$. Total momentum is conserved to find the ball's final velocity.

• Challenge Problem 20: A $0.50\,kg$ ball at $6.0\,m/s$ hits a $1.00\,kg$ ball moving opposite at $12.0\,m/s$. The first ball bounces back at $14\,m/s$.

Recoil

• Definition: The backward motion of an object (like Skater C) caused by an internal force pushing another object (Skater D) forward.

• Momentum Balance: If two skaters start at rest, the total initial momentum is zero. After the push, their momenta must still sum to zero: $P_{Cf} + P_{Df} = 0$, meaning $P_{Cf} = -P_{Df}$.

• Velocity Ratio: The relation is expressed as $m_C v_{Cf} = -m_D v_{Df}$. If Skater C is $68.0\,kg$ and Skater D is $45.4\,kg$, the ratio of their velocities is $1.50$. The less massive skater moves with the higher velocity.

• Evidence: In photographic analyses (Figure 11), a stationary object (like a red ball) can help define the system by providing a reference point for movement.

Propulsion in Space and Rocketry

• Mechanism: A rocket in space carries fuel and an oxidizer. When combined, hot gases are expelled at high speed from the exhaust nozzle.

• System Classification: The rocket and the chemicals constitute a closed, isolated system because the expulsion forces are internal.

• Ion Thrusters: A modern propulsion method where ions are accelerated in electric or magnetic fields. The recoil from expelling these ions in one direction moves the spacecraft in the opposite direction.

• Example Problem 4 (Astronaut Recoil):

  • Data: Astronaut + pistol mass = $84\,kg$. Gas expelled = $35\,g$ ($0.035\,kg$) at $-875\,m/s$.

  • Initial Momentum ($P_i$): $0.0\,kg\cdot m/s$.

  • Conservation: $P_{\text{astronaut}} = -P_{\text{gas}}$.

  • Calculation: $v_{\text{astronaut}} = -\frac{(0.035\,kg)(-875\,m/s)}{84\,kg} = +0.36\,m/s$.

Two-Dimensional Collisions

• Conservation in Dimensions: Momentum is conserved in a closed, isolated system regardless of the directions of particles. This is analyzed by breaking vectors into components ($x$ and $y$).

• Vector Components:

  • $P_{ix} = P_{fx}$ and $P_{iy} = P_{fy}$.

  • Example: For billiard balls, if the y-component of initial momentum is zero, the final y-components must sum to zero: $P_{Cy} + P_{Dy} = 0$.

• Example Problem 5 (Perpendicular Collision):

  • Car C: $1325\,kg$, moving North at $27.0\,m/s$. ($P_{Ci} = 3.58 \times 10^4\,kg \cdot m/s$ North).

  • Car D: $2165\,kg$, moving East at $11.0\,m/s$. ($P_{Di} = 2.38 \times 10^4\,kg \cdot m/s$ East).

  • Resulting Vector: $p_f = \sqrt{p_{Ci}^2 + p_{Di}^2} = 4.30 \times 10^4\,kg \cdot m/s$.

  • Angle: $\theta = \tan^{-1}\left(\frac{3.58 \times 10^4}{2.38 \times 10^4}\right) = 56.4^\circ$ (North of East).

  • Final Speed: $v_f = \frac{p_f}{m_C + m_D} = \frac{4.30 \times 10^4}{1325 + 2165} = 12.3\,m/s$.

Conservation of Angular Momentum

• The Law: If no net external torque acts on a closed system, the angular momentum ($L$) remains unchanged: $L_i = L_f$.

• Ice Skater Example: When a skater pulls in their arms, their moment of inertia ($I$) decreases. To conserve angular momentum ($L = I\omega$), the angular velocity ($\omega$) must increase ($I_i \omega_i = I_f \omega_f$).

• Precession: The change in the direction of the axis of a spinning object due to torque.

  • Spinning Tops: Gravity exerts no torque when a top is perfectly vertical (line through pivot). When tilted, gravity creates torque, causing the axis to precess (revolve away from the vertical).

  • Earth's Precession: Earth is not a perfect sphere (equatorial bulge). The Sun's gravitational pull exerts torque on this bulge, causing a precession cycle of approximately $26,000$ years.

Gyroscopes and Practical Applications

• Gyroscope: A wheel/disk spinning rapidly around one axis, free to rotate around others. Its Large angular momentum resists direction changes unless a torque is applied.

• Uses: In airplanes, submarines, and spacecraft to maintain a stable reference direction. Gyroscopic compasses work regardless of level surfacing.

• Sports Applications:

  • Football: Quarterbacks throw spirals to use the gyroscope effect. Spinning the ball around its long axis allows it to maintain its pointed end forward, reducing air resistance and preventing tumbling.

  • Flying Disks: Spin stabilizes flight, preventing wobbling.

  • Yo-yos: Fast rotation keeps the yo-yo in a single plane.

Questions & Discussion

• Physics Challenge - Colliding Cars: A friend in a $1265\,kg$ car moving North was hit by a $925\,kg$ car moving West. They stuck together and slid $23.1\,m$ at $42^\circ$ North of West. Using momentum conservation and kinetic friction ($\mu_k = 0.65$), one can determine if the friend was speeding (limit $22\,m/s$).

• Check Your Progress Highlights:

  • Q30: Momentum is conserved in a tennis serve only if the system includes the racket and the ball (ignoring the player's external force).

  • Q31: The vertical momentum of a pole-vaulter comes from the conversion of horizontal kinetic energy and work done by the athlete, using the pole as a tool to redirect force.

  • Q32: If two soccer players collide and come to rest, their initial momenta were equal in magnitude but opposite in direction ($P_{total} = 0$).

  • Q33: When catching a ball on a skateboard vs. ground: On a skateboard, the system is you + ball + board (isolated horizontally), so you move. On the ground, the system includes the Earth, which has such high mass that its change in velocity is undetectable.