IBDP Physics - A Space time and motion quick revision

Understanding Motion Basics

• Position, Velocity, and Acceleration: Motion is described by how an object’s position changes over time.

  • Position ($\vec{r}$): Where an object is located in space.
  • Velocity ($\vec{v}$): The rate of change of position. It’s a vector quantity (has magnitude and direction). $\vec{v} = \frac{d\vec{r}}{dt}$
  • Acceleration ($\vec{a}$): The rate of change of velocity. It’s also a vector quantity. $\vec{a} = \frac{d\vec{v}}{dt}$

• Displacement vs. Distance:

  • Displacement ($\Delta \vec{r}$): The change in position of an object (a vector). It’s the shortest distance between the initial and final points.
  • Distance (d): The total path length traveled by an object (a scalar).

• Instantaneous vs. Average:

  • Instantaneous Velocity/Speed: The velocity/speed at a specific moment in time.
  • Average Velocity: Total displacement divided by the total time taken. $\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}$
  • Average Speed: Total distance traveled divided by the total time taken.

Uniformly Accelerated Motion Equations

These equations apply when the acceleration is constant:

• $v = u + at$
• $s = ut + \frac{1}{2}at^2$
• $v^2 = u^2 + 2as$
• $s = \frac{u+v}{2}t$

Where:

• $s$ = displacement
• $u$ = initial velocity
• $v$ = final velocity
• $a$ = uniform acceleration
• $t$ = time

Types of Motion

• Uniform Motion: Velocity is constant (acceleration is zero).
• Non-Uniform Motion: Velocity is changing (acceleration is non-zero).

Motion with Fluid Resistance

• In real-world scenarios, fluids (like air and water) exert a resistance force on moving objects.
• This force depends on factors like the object’s shape, size, and speed, as well as the fluid’s properties.
• Fluid resistance complicates the equations of motion, making acceleration non-constant.

Projectile Motion

• Projectile motion is the motion of an object thrown or projected into the air, subject only to gravity (ignoring air resistance in simplified cases).
• Key Concepts (without fluid resistance):
  • The trajectory is a parabola.
  • Horizontal and vertical motions are independent.
  • Horizontal velocity remains constant.
  • Vertical motion is uniformly accelerated due to gravity ($g \approx 9.8 \, m/s^2$ downwards).
• Important Parameters:
  • Time of flight: The total time the projectile is in the air.
  • Trajectory: The path followed by the projectile.
  • Velocity: The velocity of the projectile at any point (has horizontal and vertical components).
  • Acceleration: Constant and downwards ($g$).
  • Range: The horizontal distance covered by the projectile.
  • Terminal Speed: The constant speed that a freely falling object eventually reaches when the force of air resistance equals the force of gravity (relevant when considering fluid resistance).

Projectile Motion (Simplified Focus)

• We often analyze projectile motion by considering constant gravitational acceleration near the Earth’s surface.
• Problems usually involve projectiles launched horizontally, at angles above the horizontal, or at angles below the horizontal.

Quick Revision Notes: Forces and Momentum

Newton’s Laws of Motion

• Newton’s First Law (Law of Inertia): An object at rest stays at rest, and an object in motion stays in motion with the same velocity unless acted upon by a net force.
• Newton’s Second Law: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. $\vec{F}_{net} = m\vec{a}$
• Newton’s Third Law: For every action, there is an equal and opposite reaction. When one object exerts a force on a second object, the second object simultaneously exerts a force equal in magnitude and opposite in direction on the first object.

Forces as Interactions

Forces arise from interactions between objects. These interactions can be contact forces or field forces.

Free-Body Diagrams

A free-body diagram is a visual representation of all the external forces acting on an object. It helps in analyzing the net force and applying Newton’s laws.

Common Contact Forces

• Normal Force ($F_N$): The component of the contact force exerted by a surface on an object that is perpendicular to the surface.
• Friction ($F_f$): A force that opposes motion between surfaces in contact.
  • Static Friction ($F_{fs}$): Prevents an object from starting to move. $F_{fs} \leq \mu_s F_N$, where $\mu_s$ is the coefficient of static friction.
  • Kinetic Friction ($F_{fk}$): Acts on a moving object. $F_{fk} = \mu_k F_N$, where $\mu_k$ is the coefficient of kinetic friction. Typically, $\mu_k < \mu_s$.
• Tension ($T$): The force exerted by a stretched string, rope, or cable on an object to which it is attached.
• Elastic Restoring Force (Spring Force, $F_s$): The force exerted by a spring that tends to restore it to its equilibrium position. Given by Hooke’s Law: $F_s = -kx$, where $k$ is the spring constant and $x$ is the displacement from equilibrium.
• Viscous Drag Force ($F_d$): A resistive force exerted by a fluid on an object moving through it. For a small sphere moving slowly, Stokes’ Law applies: $F_d = 6\pi \eta r v$, where $\eta$ is the fluid viscosity, $r$ is the sphere’s radius, and $v$ is the sphere’s velocity.
• Buoyancy ($F_B$): The upward force exerted by a fluid that opposes the weight of an immersed object. $F_B = \rho V g$, where $\rho$ is the fluid density, $V$ is the volume of the fluid displaced, and $g$ is the acceleration due to gravity.

Common Field Forces

• Gravitational Force ($F_g$): The force of attraction between objects with mass. Near the Earth’s surface, $F_g = mg$, where $m$ is the mass and $g$ is the acceleration due to gravity.
• Electric Force ($F_E$): The force between electrically charged objects.
• Magnetic Force ($F_M$): The force exerted on moving electric charges and magnetic materials in a magnetic field.

Linear Momentum

• Linear Momentum ($\vec{p}$): A measure of the mass in motion. It is a vector quantity defined as the product of an object’s mass ($m$) and its velocity ($\vec{v}$). $\vec{p} = m\vec{v}$
• Conservation of Linear Momentum: In a closed system (no external forces), the total linear momentum remains constant. $\sum \vec{p}_{initial} = \sum \vec{p}_{final}$
• Impulse ($\vec{J}$): The change in momentum of an object. It is also equal to the net force acting on the object multiplied by the time interval over which the force acts. $\vec{J} = \Delta \vec{p} = \vec{F}_{net} \Delta t$

Newton’s Second Law in Terms of Momentum

Newton’s second law can also be expressed as the rate of change of momentum: $\vec{F}_{net} = \frac{d\vec{p}}{dt}$ If mass is constant, this reduces to $\vec{F}_{net} = m\vec{a}$.

Collisions

• Elastic Collision: A collision in which both kinetic energy and momentum are conserved.
• Inelastic Collision: A collision in which momentum is conserved, but kinetic energy is not. Some kinetic energy is typically lost as heat or sound.
• Explosions: A process where a stationary object breaks into multiple moving parts. Momentum is conserved (the initial momentum is zero).

Energy Considerations in Collisions

• In elastic collisions, the total kinetic energy before the collision equals the total kinetic energy after the collision.
• In inelastic collisions, the total kinetic energy after the collision is less than the total kinetic energy before the collision.

Circular Motion

• Uniform Circular Motion: Motion along a circular path with a constant speed. Although the speed is constant, the velocity is not because the direction is continuously changing.
• Centripetal Acceleration ($a_c$): The acceleration directed towards the center of the circular path that causes the change in direction of the velocity. $a_c = \frac{v^2}{r} = \omega^2 r$ where $v$ is the speed, $r$ is the radius of the circular path, and $\omega$ is the angular speed.
• Centripetal Force ($F_c$): The net force directed towards the center of the circular path that causes the centripetal acceleration. $F_c = ma_c = \frac{mv^2}{r} = m\omega^2 r$
• Angular Velocity ($\omega$): The rate of change of angular displacement. $\omega = \frac{v}{r}$
• Relationship between Linear and Angular Velocity: $v = \omega r$.
• Angular Displacement ($\theta$): The angle through which a point or line has been rotated in a specified direction about a specified axis.
• Period ($T$): The time taken for one complete revolution. $\omega = \frac{2\pi}{T}$.
• Frequency ($f$): The number of revolutions per unit time. $f = \frac{1}{T}$.

Angular Velocity and Linear Velocity

The linear velocity ($\vec{v}$) of a point moving along a circular trajectory is related to the angular velocity ($\vec{\omega}$) by: $v = \omega r$ where $r$ is the radius of the circular path. The direction of the linear velocity is tangential to the circle.

Quick Revision Notes: Energy, Work, and Power

Conservation of Energy

The principle of the conservation of energy states that energy cannot be created or destroyed; it can only be transformed from one form to another or transferred between objects or systems. The total energy of an isolated system remains constant.

Work and Energy Transfer

Work done by a force on an object is equivalent to the transfer of energy to or from that object. If work is done on a system, its energy increases; if work is done by a system, its energy decreases.

Sankey Diagrams

Sankey diagrams are flow diagrams that represent energy transfers in a system. The width of the arrows is proportional to the amount of energy being transferred or transformed. They visually show the useful energy output and the wasted energy.

Work Done by a Constant Force

The work ($W$) done by a constant force ($\vec{F}$) on an object as it undergoes a displacement ($\vec{s}$) is given by the dot product of the force and the displacement vectors: $W = \vec{F} \cdot \vec{s} = Fs \cos \theta$ where $F$ is the magnitude of the force, $s$ is the magnitude of the displacement, and $\theta$ is the angle between the force and displacement vectors. Only the component of the force along the line of displacement ($F \cos \theta$) does work.

Work-Energy Theorem

The work done by the resultant force on a system is equal to the change in the total energy of the system. This is known as the work-energy theorem.

Mechanical Energy

Mechanical energy ($E_{mech}$) is the sum of the kinetic energy ($E_k$) and potential energy ($E_p$) of a system. For systems near the Earth’s surface, potential energy often includes gravitational potential energy ($E_g$) and elastic potential energy ($E_{elastic}$). $E_{mech} = E_k + E_p = E_k + E_g + E_{elastic}$

Conservation of Mechanical Energy

In the absence of non-conservative forces like friction and air resistance, the total mechanical energy of a system remains constant. Energy can be transformed between kinetic and potential forms, but the total sum does not change.

Forms of Mechanical Energy

• Kinetic Energy ($E_k$) of Translational Motion: The energy an object possesses due to its motion. $E_k = \frac{1}{2}mv^2 = \frac{p^2}{2m}$ where $m$ is the mass and $v$ is the speed of the object, and $p$ is the linear momentum.

• Gravitational Potential Energy ($\Delta E_g$): The energy an object possesses due to its position in a gravitational field relative to a reference point. Near the Earth’s surface, the change in gravitational potential energy is: $\Delta E_g = mg\Delta h$ where $m$ is the mass, $g$ is the acceleration due to gravity, and $\Delta h$ is the change in height.

• Elastic Potential Energy ($E_{elastic}$): The energy stored in an elastic material (like a spring) when it is stretched or compressed. $E_{elastic} = \frac{1}{2}k(\Delta x)^2$ where $k$ is the spring constant and $\Delta x$ is the displacement from the equilibrium position.

Power

Power ($P$) is the rate at which work is done or the rate at which energy is transferred. $P = \frac{\Delta W}{\Delta t} = \frac{\Delta E}{\Delta t}$ For an object moving with velocity $v$ under the influence of a force $F$ in the direction of motion, power can also be expressed as: $P = Fv$

Efficiency

Efficiency ($\eta$) is the ratio of useful energy output to the total energy input, or the ratio of useful power output to the total power input. It is a dimensionless quantity, often expressed as a percentage. $\eta = \frac{\text{useful work out}}{\text{total work in}} = \frac{\text{useful power out}}{\text{total power in}}$ Efficiency is always less than or equal to 1 (or 100%) due to energy losses (e.g., as heat).

Energy Density of Fuel Sources

Energy density refers to the amount of energy that can be stored in a given mass or volume of a fuel. It is typically measured in units like joules per kilogram (J/kg) or joules per liter (J/L). Different fuel sources have different energy densities, which affects their suitability for various applications.

Quick Revision Notes: Rotational Motion and Torque

Torque

• Definition: Torque ($\tau$) is a rotational equivalent of linear force. It is a measure of how much a force acting on an object causes that object to rotate.
• Equation: The magnitude of the torque is given by: $\tau = r F \sin \theta$ where $r$ is the magnitude of the position vector from the axis of rotation to the point where the force is applied, $F$ is the magnitude of the force, and $\theta$ is the angle between the force vector and the position vector.
• Vector Form: Torque is a vector quantity: $\vec{\tau} = \vec{r} \times \vec{F}$. The direction of the torque is perpendicular to both $\vec{r}$ and $\vec{F}$, determined by the right-hand rule.
• Rotational Equilibrium: For a body to be in rotational equilibrium, the net torque acting on it must be zero ($\sum \tau = 0$).

Angular Kinematics

These equations describe rotational motion with constant angular acceleration:

• $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$
• $\omega = \omega_0 + \alpha t$
• $\omega^2 = \omega_0^2 + 2\alpha \Delta \theta$
• $\Delta \theta = \frac{\omega_0 + \omega}{2} t$

Where:

• $\theta$ = angular displacement
• $\theta_0$ = initial angular displacement
• $\omega$ = final angular velocity
• $\omega_0$ = initial angular velocity
• $\alpha$ = angular acceleration
• $t$ = time

Moment of Inertia

• Definition: Moment of inertia ($I$) is the rotational equivalent of mass. It is a measure of an object’s resistance to changes in its rotational motion.
• Dependence: The moment of inertia of an extended body depends on the distribution of its mass relative to the axis of rotation. Objects with mass concentrated farther from the axis of rotation have a larger moment of inertia.
• For a system of point masses: $I = \sum m_i r_i^2$, where $m_i$ is the mass of each point and $r_i$ is its distance from the axis of rotation.

Newton’s Second Law for Rotation

Newton’s second law for rotational motion states that the net torque ($\tau_{net}$) acting on an object is directly proportional to its angular acceleration ($\alpha$) and its moment of inertia ($I$): $\tau_{net} = I \alpha$

Angular Momentum

• Definition: Angular momentum ($L$) is the rotational equivalent of linear momentum. It is a measure of an object’s tendency to continue rotating.
• For an extended body rotating with angular velocity $\omega$ about a fixed axis: $L = I \omega$
• Vector Form: $\vec{L} = I \vec{\omega}$. The direction of angular momentum is along the axis of rotation, determined by the right-hand rule.

Conservation of Angular Momentum

If the net external torque acting on a system is zero, the total angular momentum of the system remains constant. $\sum \vec{L}_{initial} = \sum \vec{L}_{final}$

Work and Kinetic Energy in Rotational Motion

• Work done by a torque: When a torque $\tau$ acts on an object causing an angular displacement $\Delta \theta$, the work done is: $W = \tau \Delta \theta$
• Kinetic energy of rotational motion ($E_{kr}$): The energy an object possesses due to its rotation. $E_{kr} = \frac{1}{2} I \omega^2 = \frac{L^2}{2I}$

Torque and Angular Impulse

The angular impulse is the change in angular momentum. It is equal to the net torque multiplied by the time interval over which the torque acts: $\Delta L = \tau \Delta t$

Vector Nature

Torque and angular momentum are vector quantities. Their direction is crucial and is typically determined using the right-hand rule. The sense (clockwise or counter-clockwise) of the torque and angular momentum should be considered in problem-solving.

Quick Revision Notes: Special Relativity

• Reference Frames: A coordinate system used to describe the motion of an object or the occurrence of an event.
• Inertial Reference Frames: Frames of reference in which Newton’s laws of motion hold true. These are non-accelerating frames.
• Galilean Relativity: The principle that the laws of motion are the same in all inertial reference frames. Velocities are additive according to the Galilean transformation.
  • Position transformation: $x' = x - vt$
  • Time transformation: $t' = t$ (absolute time)
  • Velocity addition: $u' = u - v$ (where $v$ is the relative velocity between the frames)

Postulates of Special Relativity (Einstein’s Postulates)

1. The laws of physics are the same for all observers in all inertial frames of reference. This extends the principle of Galilean relativity to all laws of physics, including electromagnetism.
2. The speed of light in a vacuum (c) is the same for all inertial observers, regardless of the motion of the light source. This postulate has profound consequences for our understanding of space and time.

Lorentz Transformation

The postulates of special relativity lead to the Lorentz transformation equations, which relate the space and time coordinates of an event as measured by two observers in inertial frames moving relative to each other with a constant velocity $v$ along the x-axis:

• $x' = \gamma (x - vt)$
• $t' = \gamma (t - \frac{vx}{c^2})$

where $\gamma$ is the Lorentz factor: $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$

Relativistic Velocity Addition

The Lorentz transformation leads to the relativistic velocity addition equation:

• $u' = \frac{u - v}{1 - \frac{uv}{c^2}}$ (for motion along the x-axis)

This equation shows that velocities do not simply add linearly at relativistic speeds, and the relative velocity will never exceed $c$.

Space-Time Interval

• The space-time interval ($\Delta s$) between two events is an invariant quantity, meaning it has the same value for all inertial observers.
• $(\Delta s)^2 = (c\Delta t)^2 - (\Delta x)^2 - (\Delta y)^2 - (\Delta z)^2$
• For motion along one spatial dimension: $(\Delta s)^2 = (c\Delta t)^2 - (\Delta x)^2$

Proper Time and Proper Length

• Proper Time ($\Delta \tau$ or $\Delta t_0$): The time interval between two events measured by an observer who is at rest with respect to both events (they occur at the same spatial location in that observer’s frame).
  • $\Delta t = \gamma \Delta t_0$ (Time dilation: moving clocks run slower)
• Proper Length ($L_0$): The length of an object measured by an observer who is at rest with respect to the object.
• Length Contraction ($L$): The length of an object measured by an observer who is moving relative to the object along the direction of the length measurement. The length $L$ is shorter than the proper length $L_0$.
  • $L = \frac{L_0}{\gamma}$

Relativity of Simultaneity

Two events that are simultaneous in one inertial frame of reference may not be simultaneous in another inertial frame that is moving relative to the first. Simultaneity is relative.

Space-Time Diagrams

• Diagrams that plot time (usually $ct$) on one axis and one or more spatial dimensions on the other.
• Worldline: The path of an object through space-time.
• The angle that the worldline of a moving particle makes with the time axis is related to the particle’s velocity. $\tan \theta = \frac{v}{c}$.

Muon Decay Experiment

The increased lifetime of muons traveling at relativistic speeds, as observed from Earth compared to their proper lifetime, provides experimental evidence for time dilation. The fact that muons created high in the atmosphere can reach the Earth’s surface is also explained by length contraction in the muon’s frame of reference (the distance to the surface is contracted).