Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Classical mechanics → dynamics ↓


projectile motion


Projectile motion is a type of motion experienced by an object that is projected into the air and that is affected by gravity. It is an important topic in physics that incorporates the concepts of both kinematics and dynamics, and it provides essential information about the motion of objects.

When we talk about projectile motion, we are considering objects that are initially pushed into the air, and which move along a curved trajectory under the force of gravity only. No other forces act on these objects, assuming we neglect air resistance and other forces. Classical examples include a ball thrown into the air, a cannonball fired from a cannon, or a stone thrown from a cliff.

Core beliefs

To simplify the study of projectile motion, we usually make some assumptions:

  • The only significant force acting on a projectile is gravity.
  • Air resistance is negligible.
  • The projectile travels a relatively short distance so the curvature of the Earth can be neglected.
  • The acceleration due to gravity g is constant and acts downward with a magnitude of about 9.81 m/s².

Components of projectile motion

Projectile motion can be analyzed by dividing it into two components: horizontal and vertical. The key to understanding projectile motion is to understand that these two components are independent of each other except for the time of flight. Here, we will describe these components.

Horizontal speed

The horizontal component of the projectile's motion is controlled by the initial velocity and time. In the absence of air resistance, the horizontal velocity v_x remains constant. It is described by the formula:

v_x = v_i * cos(θ)

Where v_i is the initial velocity and θ is the projection angle. Thus the horizontal displacement x can be determined using:

x = v_x * t

Vertical speed

The vertical motion is subject to gravitational acceleration, and the initial vertical velocity v_y is given by:

v_y = v_i * sin(θ)

However, the vertical speed changes continuously because of the acceleration due to gravity. The equations for vertical velocity and vertical displacement height y are:

v_y = v_i * sin(θ) - g * t
y = v_i * sin(θ) * t - (1/2) * g * t²

Flight time

The time the projectile is in the air, or "time of flight", is determined by the vertical component of the speed. For a projectile launched and landed at the same vertical level, the total time of flight T is given by:

T = (2 * v_i * sin(θ)) / g

Maximum height

The maximum height H reached by the projectile is also determined by the vertical component of the initial velocity:

h = (v_i² * sin²(θ)) / (2 * g)

Range of the projectile

The horizontal distance traveled by the projectile, known as its range R, is given by:

r = (v_i² * sin(2θ)) / g

This formula assumes that the projectile will fall at the same vertical level from which it was released.

Illustration of projectile motion

Let's visualize projectile motion with a simple illustration, showing the trajectory and key points:

Peak (maximum height) launch Effect

Real-world examples and problems

Let's take a look at some common examples of projectile motion and see how we can apply the formulas we've derived:

Example 1: Kicking a football

Imagine that a football is kicked at an angle of 30° with an initial velocity of 20 m/s. Let's find the time, maximum height and distance of the football's flight.

  • Flight time:
    Given, v_i = 20 m/s, θ = 30°, g = 9.81 m/s²
    t = (2 * 20 * sin(30)) / 9.81 ≈ 2.04 sec
    Therefore, the football remains in the air for approximately 2.04 seconds.
  • Maximum height:
    H = (20² * sin²(30)) / (2 * 9.81) ≈ 5.10 m
    The maximum height of the football is approximately 5.10 meters.
  • Category:
    r = (20² * sin(60)) / 9.81 ≈ 34.64 m
    Therefore, the football covers a horizontal distance of approximately 34.64 meters.

Example 2: Basketball throw

Suppose a player is throwing a basketball at an initial speed of 15 m/s and at an angle of 45°. Find the range of this basketball throw.

Range, R = (v_i² * sin(2θ)) / g
r = (15² * sin(90)) / 9.81 ≈ 22.94 m

If we assume there is no air resistance the basketball will fall approximately 22.94 meters from the point where it was thrown.

Complex ideas for advanced study

In real-world scenarios, factors such as air resistance, wind, spin, and the shape and mass of the object can affect projectile motion. Such considerations make projectile motion much more complex and require more sophisticated mathematical models beyond the basics we've touched on here.

For example, when air resistance is not negligible, it can be modeled as a force proportional to the velocity or the square of the velocity, depending on the speed and characteristics of the projectile. This will lead to differential equations that will require numerical methods or approximations to solve.

Conclusion

An understanding of simple projectile motion is foundational to more complex physics topics. Engineers, scientists, and professionals in fields as diverse as sports, space exploration, and defense often use the principles of projectile motion. By understanding the fundamentals of how objects move in space under the influence of gravity, we can more accurately predict and model real-world phenomena.


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projectile motion

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