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 ↓


Projectile motion


Projectile motion is an important topic in the study of dynamics, a branch of mechanics that deals with the motion of objects without considering the forces that cause this motion. Understanding projectile motion involves analyzing the path that an object follows when it is thrown, propelled, or projected into space. A classic example is a ball thrown into the air at an angle.

Projectile motion occurs when an object is launched into the air at an angle. The object will follow a curved path due to the effect of gravity, which acts downward. This path is often called its trajectory.

Components of projectile motion

Projectile motion can be analyzed by breaking it down into two components: horizontal and vertical motion. The key to understanding projectile motion is to recognize that these two motion components act independently of one another except for the time component. There are no forces acting horizontally (if we neglect air resistance), while gravity acts vertically.

Horizontal speed

In the horizontal direction, there is no force acting on the projectile (assuming that air resistance is negligible), and thus it moves forward at a constant velocity. This can be described by the equation:

( v_x = v_{0x} ) ( x = v_{0x} cdot t )

Where:

  • ( v_x ) is the horizontal velocity.
  • ( v_{0x} ) is the initial horizontal velocity.
  • ( x ) is the horizontal distance traveled.
  • ( t ) is the elapsed time.

Vertical speed

In the vertical direction, the projectile is affected by gravity. This means that it will accelerate downward at a constant rate of (g), which is approximately (9.81,m/s^2) on Earth. The equations for vertical motion include:

( v_y = v_{0y} - g cdot t ) ( y = v_{0y} cdot t - frac{1}{2} g cdot t^2 )

Where:

  • ( v_y ) is the vertical velocity.
  • ( v_{0y} ) is the initial vertical velocity.
  • ( y ) is the vertical displacement.
  • ( g ) is the acceleration due to gravity.

General equation for projectile motion

The two components, horizontal and vertical, can be combined into a complete description of projectile motion. The initial velocity ( v_0 ) at which the object is projected can be split into its components ( v_{0x} ) and ( v_{0y} ) using the projection angle ( theta ):

( v_{0x} = v_0 cdot cos(theta) ) ( v_{0y} = v_0 cdot sin(theta) )

These initial component velocities can be used in the previous equations of horizontal and vertical motion to describe the complete projectile motion.

Path of the projectile

A projectile follows a parabolic trajectory because the horizontal motion is uniform and the vertical motion is uniformly accelerated. Mathematically, the path or trajectory can be described as:

( y = x cdot tan(theta) - frac{g}{2 cdot v_{0x}^2} cdot x^2 )

This equation represents the trajectory of a projectile in the absence of air resistance.

Example of projectile motion

Let us consider an example:

Suppose you throw a ball with an initial velocity of ( 20 , m/s ) at an angle of ( 30^circ ) to the horizontal. Determine how far it travels before hitting the ground.

Step-by-step solution:

  1. Calculate the initial velocity components: 
    ( v_{0x} = 20 cdot cos(30^circ) = 20 cdot (√3/2) approx 17.32 , m/s ) ( v_{0y} = 20 cdot sin(30^circ) = 20 cdot (1/2) = 10 , m/s )
  2. Find the flight time:

    Use the vertical momentum equation where the final vertical position is 0 (when it hits the ground):

    ( y = v_{0y} cdot t - frac{1}{2} g cdot t^2 = 0 ) ( 10 cdot t - frac{1}{2} cdot 9.81 cdot t^2 = 0 ) ( t cdot (10 - frac{1}{2} cdot 9.81 cdot t) = 0 )

    Solving this for ( t ) is simple:

    ( t = 0 ) or ( t = frac{10}{4.905} approx 2.04 , s )

    Since ( t = 0 ) is the time when the ball was initially launched, we take ( t = 2.04 , s ).

  3. Calculate the horizontal distance (range):

    Now calculate the distance, using the time of flight:

    ( x = v_{0x} cdot t = 17.32 cdot 2.04 approx 35.32 , m )

The ball travels a distance of approximately ( 35.32 , meters ) before falling to the ground.

Visual concept

Let us look at the motion of a projectile through a simple graphical representation of its components and trajectory.

Projection Point Top Impact Point Trajectory

In the above illustration:

  • The black lines indicate the x-axis (horizontal projection) and y-axis (vertical elevation).
  • The blue curve shows the trajectory of the projectile.
  • The red circles represent the launch and impact points, while the green circle represents the apex, or highest point of the trajectory.

The physics behind projectile motion

The physics behind projectile motion involves the interaction between the forces and velocities acting on the projectile. Here's a closer look:

Initial velocity and angle

The initial velocity and launch angle are important in determining the range and height of the projectile. By adjusting the launch angle and speed, the trajectory of the projectile can be manipulated:

  • A launch angle of ( 45^circ ) generally maximizes range when air resistance is ignored.
  • A steeper angle results in a shorter distance, but a higher maximum height.
  • The shallower angle results in a greater distance, but a lower maximum height.

Freedom of motion

The independence of the horizontal and vertical components simplifies the analysis of projectile motion. This theory states:

  • If air resistance is ignored the horizontal velocity remains constant.
  • Once the projectile is in motion, the vertical motion is entirely affected by gravity.

Factors affecting projectile motion

When solving ideal projectile motion problems, several real-world factors can affect the motion:

Air resistance

Air resistance acts opposite to the direction of motion and can significantly affect both horizontal speed and maximum altitude:

  • This reduces the range and altitude of the projectile.
  • In the presence of air resistance the trajectory becomes more complex and less predictable.

Spin and lift

Some projectiles, such as balls or Frisbees, can have spin. The spin can create lift, which can change the path of the projectile:

  • The Magnus effect causes projectiles with spin to experience lift perpendicular to their velocity.
  • Spin can stabilize the projectile, affecting accuracy and precision in sports such as golf or football.

Advanced problem example

Let's look at a more complex example involving projectile motion.

A cannonball is fired from the edge of a cliff ( 100 , m ) above the sea level, at an angle of ( 37^circ ) to the horizontal, with an initial velocity of ( 50 , m/s ). At what distance from the base of the cliff will it fall into the sea?

Solution steps:

  1. Determine the initial velocity components: 
    ( v_{0x} = 50 cdot cos(37^circ) = 50 cdot 0.7986 approx 39.93 , m/s ) ( v_{0y} = 50 cdot sin(37^circ) = 50 cdot 0.6018 approx 30.09 , m/s )
  2. Find the flight time:

    Use the vertical momentum equation where the final vertical position takes into account the height of the rock:

    ( y = v_{0y} cdot t - frac{1}{2} g cdot t^2 = -100 ) ( 30.09 cdot t - frac{1}{2} cdot 9.81 cdot t^2 = -100 )

    Solving this quadratic equation gives:

    ( t approx 8.71 , s )
  3. Calculate the horizontal distance (range) when the cannonball hits the sea: 
    ( x = v_{0x} cdot t = 39.93 cdot 8.71 approx 347.72 , m )

The cannonball will fall into the sea about ( 347.72 , meters ) above the base of the cliff.

Conclusion

Projectile motion provides an elegant description of any object projected into space, subject only to gravity. Understanding how to break down the various components of motion and how to calculate them is essential for predicting trajectories in physics and engineering applications.

The study of projectile motion equips students with the skills needed to solve real-world problems, enhance their understanding of physics concepts and apply these principles to practical scenarios.


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Projectile motion

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