# Lecture 9: Linear Momentum and Collisions ## Physics I — Mechanics & Thermodynamics --- ## Learning Objectives By the end of this lecture, students will be able to: 1. Define linear momentum and relate it to force via the impulse-momentum theorem 2. Calculate impulse from force-time graphs and equations 3. Apply conservation of momentum to isolated systems 4. Distinguish between elastic, inelastic, and perfectly inelastic collisions 5. Solve one-dimensional and two-dimensional collision problems 6. Locate the center of mass of a system and describe its motion --- ## Lecture Content ### I. Linear Momentum The **linear momentum** of a particle is defined as **p** = m**v**. It is a vector quantity pointing in the same direction as the velocity, with SI units of kg m/s. Momentum provides a measure of how hard it is to stop a moving object: a large, slow truck and a small, fast bullet can have comparable momenta. Newton's second law can be expressed more generally in terms of momentum as **F**_net = d**p**/dt. For an object of constant mass, this reduces to the familiar form **F**_net = m**a**. The momentum formulation is more general because it also applies to situations where the mass changes, such as rocket propulsion. ### II. Impulse and the Impulse-Momentum Theorem **Impulse** (J) is defined as the product of force and the time interval over which it acts: **J** = integral of **F** dt (from t_i to t_f). For a constant force, this simplifies to **J** = **F** Delta t. The **impulse-momentum theorem** states that the impulse equals the change in momentum: **J** = Delta **p** = **p**_f - **p**_i = m**v**_f - m**v**_i. Graphically, the impulse is the **area under the force vs. time curve**. This theorem has profound practical applications. Airbags and crumple zones in cars increase the collision time Delta t, thereby reducing the average force for the same impulse. Similarly, catching a ball by following through with the hands increases the contact time and reduces the impact force. A bouncing ball experiences greater impulse (and therefore greater force) than one that simply stops, because the change in momentum is larger when the ball reverses direction. Panel A: A force vs. time graph for a brief collision (e.g., a ball hitting a wall). The curve rises sharply to a peak F_max, then falls back to zero. The shaded area under the curve is labeled "Impulse J = area." A horizontal dashed line at the average force F_avg is shown, with a rectangle of the same area. Panel B: Two scenarios — a car hitting a rigid wall (short Delta t, high F) vs. a car with a crumple zone (longer Delta t, lower F). Both have the same impulse (same Delta p), but the force profiles differ dramatically. ### III. Conservation of Linear Momentum For an **isolated system** where no external forces act (or where external forces sum to zero), the total momentum is conserved: d**p**_total / dt = 0, so **p**_total = constant. In component form, m_1 **v**_1i + m_2 **v**_2i = m_1 **v**_1f + m_2 **v**_2f. This conservation holds in **every direction independently**: p_x is conserved if there is no net external force in the x-direction, and p_y is conserved if there is no net external force in the y-direction. Momentum conservation is one of the most fundamental laws in physics. It applies even when the forces between objects are complex and unknown, making it indispensable for analyzing explosions and collisions. ### IV. Types of Collisions An **elastic collision** conserves both momentum and kinetic energy. This is the ideal case and approximately describes billiard ball collisions and atomic-scale interactions. An **inelastic collision** conserves momentum but not kinetic energy; some kinetic energy is converted to thermal energy, sound, and deformation. Most real-world collisions are inelastic. A **perfectly inelastic collision** is one in which the objects **stick together** after impact, producing the maximum possible kinetic energy loss for given initial conditions. The final velocity is v_f = (m_1 v_1i + m_2 v_2i) / (m_1 + m_2), and the kinetic energy lost is Delta K = (1/2)(m_1 m_2 / (m_1 + m_2))(v_1i - v_2i)^2. ### V. Elastic Collisions in 1D For a one-dimensional elastic collision between m_1 (moving) and m_2 (initially at rest), the final velocities are v_1f = [(m_1 - m_2) / (m_1 + m_2)] v_1i and v_2f = [2m_1 / (m_1 + m_2)] v_1i. Several special cases illuminate the physics. When the masses are **equal** (m_1 = m_2), the first object stops and the second moves off with the original velocity, so the objects exchange velocities. When a **heavy object hits a light one** (m_1 >> m_2), the heavy object barely slows down while the light object flies off at approximately twice the incoming speed. When a **light object hits a heavy one** (m_1 << m_2), the light object bounces back at nearly its original speed while the heavy object barely moves. For the general case where both objects are initially moving, use both conservation of momentum and conservation of kinetic energy, or equivalently apply the relative velocity relation: v_1i - v_2i = -(v_1f - v_2f). This elegant result states that the relative velocity of approach equals the relative velocity of separation in elastic collisions. ### VI. Collisions in Two Dimensions In two-dimensional collisions, momentum is conserved component-by-component: in the x-direction, m_1 v_{1ix} + m_2 v_{2ix} = m_1 v_{1fx} + m_2 v_{2fx}, and in the y-direction, m_1 v_{1iy} + m_2 v_{2iy} = m_1 v_{1fy} + m_2 v_{2fy}. For an elastic 2D collision with equal masses and one initially at rest, the two objects move off at 90 degrees to each other, a result familiar from billiards. The general strategy is to resolve all velocities into components, apply the conservation laws, and solve the resulting system of equations. A 2D collision diagram viewed from above. A ball m_1 approaches a stationary ball m_2 along the x-axis. After the collision, m_1 moves at angle theta_1 above the x-axis and m_2 moves at angle theta_2 below the x-axis. Momentum vectors before and after are shown. The vector triangle for conservation of momentum (p_1i = p_1f + p_2f) is drawn to the side, showing how the initial momentum vector equals the vector sum of the two final momentum vectors. ### VII. Center of Mass The **center of mass** (CM) is the mass-weighted average position of a system: x_cm = (m_1 x_1 + m_2 x_2 + ...) / (m_1 + m_2 + ...), with analogous expressions for y_cm and z_cm. For a continuous body, x_cm = (1/M) integral of x dm. The motion of the center of mass is governed by **v**_cm = **p**_total / M_total and **F**_net,ext = M_total **a**_cm. If there are no external forces, the center of mass moves with constant velocity or remains at rest. In any collision, the CM velocity is unchanged because momentum is conserved. This has a striking practical consequence: a thrown wrench may tumble through the air in a complicated way, but its center of mass follows a smooth parabolic trajectory. A dumbbell-shaped object (two balls connected by a rod) thrown into the air, shown at several moments in time. Each ball traces a complex wobbling path, but the center of mass (marked with a cross at each time step) follows a smooth parabolic trajectory. The parabolic path of the CM is drawn as a smooth dashed curve. An inset shows the formula for the CM position of a two-mass system.