Grade 11
  1. Mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two and three dimensions  1.1.3. Displacement and Distance  1.1.4. Speed and Velocity  1.1.5. Acceleration and deceleration  1.1.6. Graphical analysis of motion  1.1.7. Equations of motion (advanced derivations)  1.1.8. Free fall and terminal velocity  1.1.9. Projectile motion  1.1.10. Relative velocity in one and two dimensions  1.1.11. Motion of a car on an inclined plane  1.2. Dynamics  1.2.1. Newton's laws of motion and their applications  1.2.2. Inertia and Newton's first law  1.2.3. Force and types of force  1.2.4. Static and kinetic friction  1.2.5. Circular motion and centripetal acceleration  1.2.6. Non-uniform circular motion  1.2.7. Banking of roads and curves  1.2.8. Momentum and Impulse  1.2.9. Conservation of momentum in one and two dimensions  1.2.10. Applications of Newton's Third Law  1.3. Work, Energy and Power  1.3.1. Work done by a constant and variable force  1.3.2. Work–energy theorem  1.3.3. Kinetic and potential energy  1.3.4. Gravitational and elastic potential energy  1.3.5. Conservation of mechanical energy  1.3.6. Power and instantaneous power  1.3.7. Efficiency of machines  1.3.8. Work done by conservative and non-conservative forces  1.4. Rotational motion  1.4.1. Torque and angular acceleration  1.4.2. Rotational equilibrium  1.4.3. Moment of inertia and its applications  1.4.4. Angular momentum and its conservation  1.4.5. Kinetic energy of rolling motion and rotation  1.4.6. Parallel and Perpendicular Axis Theorem  1.4.7. Dynamics of rotational motion
  2. Gravitational force  2.1. Universal gravitation  2.1.1. Newton's law of universal gravitation  2.1.2. Gravitational field and potential  2.1.3. Change in acceleration due to gravity  2.1.4. Kepler's laws of planetary motion  2.1.5. Orbital mechanics and satellites  2.1.6. Escape velocity and orbital velocity  2.1.7. Weightlessness and free fall  2.1.8. Gravitational potential energy and energy in orbits  2.1.9. Black holes and gravitational lensing
  3. Properties of matter  3.1. Fluid mechanics  3.1.1. Density and pressure in liquids  3.1.2. Pascal's theory and applications  3.1.3. Archimedes' Principle and Buoyancy  3.1.4. Bernoulli's equation and applications  3.1.5. Continuity equation and the Venturi effect  3.1.6. Surface tension and capillarity  3.1.7. Viscosity and Poiseuille's Law  3.1.8. Reynolds number and turbulence  3.2. Elasticity and deformation  3.2.1. Hooke's law and the stress-strain relation  3.2.2. Elastic modulus and applications  3.2.3. Deformation and yield strength of solids
  4. Thermal physics  4.1. Heat and temperature  4.1.1. Thermometric properties and temperature scale  4.1.2. Thermal expansion of solids, liquids and gases  4.1.3. Heat transfer system  4.1.4. Specific heat capacity and calorimetry  4.1.5. Latent heat and phase change  4.2. Kinetic theory of gases  4.2.1. Molecular model of gases  4.2.2. Kinetic explanation of temperature  4.2.3. Ideal Gas Equation and Deviations  4.2.4. Mean free path and Maxwell–Boltzmann distribution  4.3. Laws of Thermodynamics  4.3.1. Zeroth Law and Thermal Equilibrium  4.3.2. First Law and Internal Energy  4.3.3. The Second Law and Entropy  4.3.4. Carnot cycle and heat engines  4.3.5. Refrigerators and heat pumps
  5. Waves and oscillations  5.1. Simple Harmonic Motion  5.1.1. Equations of SHM  5.1.2. Energy in SHM  5.1.3. Damped and forced oscillations  5.1.4. Resonance and applications  5.1.5. Pendulum and spring-mass system  5.2. Wave motion  5.2.1. Types of mechanical waves  5.2.2. Wave motion and energy transport  5.2.3. Superposition Principle and Standing Waves  5.2.4. Reflection, Refraction and Diffraction  5.2.5. Doppler effect in sound and light  5.2.6. Pulsations and interference
  6. Electricity and Magnetism  6.1. Electrostatics  6.1.1. Electric charge and conservation  6.1.2. Coulomb's law and its applications  6.1.3. Electric field and electric flux  6.1.4. Electric potential and potential energy  6.1.5. Capacitance and Energy Stored in a Capacitor  6.2. Current Electricity  6.2.1. Ohm's law and electrical resistance  6.2.2. Resistivity and temperature dependence  6.2.3. Kirchhoff's Laws and Circuit Analysis  6.2.4. Electrical power and efficiency  6.2.5. Combination of resistors  6.3. Magnetism and Electromagnetism  6.3.1. Magnetic fields and their sources  6.3.2. Magnetic force on moving charges  6.3.3. Electromagnetic induction  6.3.4. Faraday's and Lenz's laws  6.3.5. Inductance and Transformer  6.3.6. Alternating current and its applications
  7. Optics  7.1. Reflection and Refraction  7.1.1. laws of reflection and refraction  7.1.2. Mirrors and lenses  7.1.3. Total internal reflection and optical fiber  7.2. Wave optics  7.2.1. Interference and Diffraction  7.2.2. Young's double-slit experiment  7.2.3. Polarization of light
  8. Modern Physics  8.1.1. Photoelectric effect and Einstein's theory  8.1.2. Wave–particle duality  8.1.3. Heisenberg's uncertainty principle  8.1.4. Compton Effect  8.2. Atomic and Nuclear Physics  8.2.1. Atomic model and energy levels  8.2.2. Radioactive decay and half-life  8.2.3. Nuclear Fission and Fusion
  9. Electronics and Communication  9.1. Semiconductors  9.1.1. pn junction and diode  9.1.2. Transistors and Logic Gates  9.1.3. Integrated Circuits and Applications  9.2. Communication Systems  9.2.1. Basics of Analog and Digital Communication  9.2.2. Wireless and Optical Communication  9.2.3. Modulation and Demodulation

Grade 11 → Mechanics → Dynamics ↓


Conservation of momentum in one and two dimensions


Momentum is an important concept in physics. It refers to the amount of motion of an object and depends on two variables: the mass of the object and its velocity. The mathematical representation of momentum is given by the formula:

p = m * v

Here, p is speed, m is mass, and v is the velocity of the object. The unit of speed is kilogram meter per second (kg m/s).

Momentum conservation principle

The law of conservation of momentum states that in an isolated system, the total momentum before any event is equal to the total momentum after the event, provided there is no external force acting on it. Mathematically, for two colliding objects, the principle can be expressed as:

m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final

Understanding motion in one dimension

In one-dimensional motion, objects move in a straight line. Imagine two cars colliding on a straight road. Their interaction can be represented as follows:

Car 1: m1, v1_initial Car 2: m2, v2_initial

According to conservation of momentum:

m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final

This equation helps us to calculate the unknown velocities after collision.

Example problem in one dimension

Consider two ice skaters, one has a mass of 50 kg and moves at a velocity of 4 m/s, while the other has a mass of 30 kg and moves at a velocity of -2 m/s. After colliding, they join together and move as one mass. Find their velocity after the collision.

Given:

  • m1 = 50 kg, v1_initial = 4 m/s
  • m2 = 30 kg, v2_initial = -2 m/s

Using conservation of momentum:

(50 kg * 4 m/s) + (30 kg * -2 m/s) = (50 kg + 30 kg) * v_final

Simplification:

200 kg · m/s - 60 kg · m/s = 80 kg * v_final

140 kg · m/s = 80 kg * v_final

Solve for v_final:

v_final = 140 kg · m/s / 80 kg = 1.75 m/s

After the collision the velocity of both skaters is 1.75 m/s.

Motion in two dimensions

In two-dimensional motion, objects can move along different paths while making angles. Calculating motion in two dimensions involves dividing it into two perpendicular components, often the x and y axes.

Understand with an example

Picture two pucks on an air hockey table colliding at an angle. This is shown below:

Puck A: mA Puck B: mB

To calculate the final velocities, we break down the velocities of the puck into components. Using conservation of momentum for the x and y components:

mA * vA_initial_x + mB * vB_initial_x = mA * vA_final_x + mB * vB_final_x
mA * vA_initial_y + mB * vB_initial_y = mA * vA_final_y + mB * vB_final_y

Example problem in two dimensions

Suppose puck A, with a mass of 0.5 kg, moves along the x-axis at a speed of 4 m/s, and puck B, with a mass of 0.5 kg, moves along the y-axis at a speed of 3 m/s. After the collision, puck A moves at a velocity of 3 m/s, at a 37 degree angle to the x-axis. Calculate the final velocity (both magnitude and direction) of puck B.

Given:

  • mA = 0.5 kg, vA_initial = 4 m/s
  • mB = 0.5 kg, vB_initial = 3 m/s

Breaking this down into components for Puck A after the collision:

  • vA_final_x = 3 m/s * cos(37°)
  • vA_final_y = 3 m/s * sin(37°)

Calculation of components:

vA_final_x ≈ 3 m/s * 0.8 = 2.4 m/s
vA_final_y ≈ 3 m/s * 0.6 = 1.8 m/s

For conservation of momentum:

mA * vA_initial_x + mB * 0 = mA * vA_final_x + mB * vB_final_x
0.5 kg * 4 m/s = 0.5 kg * 2.4 m/s + 0.5 kg * vB_final_x

Solve for vB_final_x:

2 kg · m/s = 1.2 kg · m/s + 0.5 kg * vB_final_x
vB_final_x ≈ 1.6 m/s

And for the y-components:

0 + 0.5 kg * 3 m/s = 0.5 kg * 1.8 m/s + 0.5 kg * vB_final_y
vB_final_y ≈ 2.4 m/s

The final velocity magnitude of puck B is calculated as:

vB_final = √((vB_final_x)² + (vB_final_y)²)
vB_final ≈ √((1.6 m/s)² + (2.4 m/s)²) ≈ 2.88 m/s

To find the direction, use the following:

θ = arctan(vB_final_y / vB_final_x)
θ ≈ arctan(2.4 / 1.6) ≈ 56.3°

Therefore, after the collision, puck B moves at a speed of about 2.88 m/s at an angle of 56.3 degrees relative to the x-axis.

Conclusion

Understanding conservation of momentum helps us predict how objects behave when they collide with each other. In real life, these principles are applied in various fields such as car crash analysis, sports dynamics, and many other areas. The key to solving such problems is to carefully break down the components and apply the law of conservation in each direction independently.


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Conservation of momentum in one and two dimensions

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