Motion, Distance, and Displacement

Motion

• Orbital speed of Earth is 30 m/s
• A particle's distance from a fixed point remaining constant does not imply it is at rest. In such cases, the particle could be in circular motion
• Motion is defined by a change in a particle's position with respect to a frame of reference (F.O.R.)
• A frame of reference is considered the observer
• Rest and motion are always relative

Distance

• Distance refers to the actual path length traveled
• Distance is a characteristic of the path
• Distance is a scalar quantity
• Distance measurement is based on the frame of reference
• The S.I. unit of distance is meters (m)

Displacement

• Displacement is the shortest distance between initial and final positions
• Displacement is a characteristic of the initial and final positions
• Displacement is a vector quantity
• Displacement is dependent on the frame of reference
• The S.I. unit of displacement is meters (m)

Problem Solving (Distance and Displacement)

• A car travels 6 km north at 45° to the east, then 4 km north at 135° to the east. The straight line joining its initial and final positions makes an angle with the east and the distance from the starting point can be calculated using trigonometry
• The displacement = √(6² + 4²) = √52 km. The direction is tan⁻¹(5)

Circular Motion Problems

• A particle moves from A to B on a semi-circle will have a displacement of 2R (where R is the radius)
• For the semi-circle, the distance covered is πR

Further insights on Distance and Displacement

• Google Maps shows distance rather than displacement
• Distance isn't always larger than displacement
• For straight-line motion without turning, distance = |displacement|; otherwise, distance > |displacement|

Speed

• Speed is the rate of change of distance with respect to time
• Speed is a scalar quantity
• Speed is frame of reference dependent
• The S.I. unit of speed is m/s

Velocity

• Velocity is the rate of change of displacement with respect to time
• Velocity is a vector quantity
• Velocity is frame of reference dependent
• The S.I. unit of velocity is m/s
• Average velocity = (net displacement) / (net time)
• Instantaneous velocity is defined as the limit of Δs/Δt as Δt approaches zero
• Average speed is greater than or equal to average velocity

Instantaneous Speed

• Instantaneous speed, or speed, is the magnitude of the instantaneous velocity
• Instantaneous speed equals the magnitude of instantaneous velocity

Average Velocity Calculation

• Average velocity can be calculated by dividing the total displacement by the total time
• Average speed is calculated by dividing the total distance by the total time

Velocity Example

• A particle moves half a distance at speed V₁ and the remaining half at V₂; the average speed is (2V₁V₂) / (V₁+V₂)

Unequal Distance and Speed Calculations

• A particle moves 1/3 of the distance with speed V₁, then the next 1/3 with V₂, and the final 1/3 with V₃. The average speed can be calculated with the formula: average speed = (3V₁V₂*V₃) / (V₁V₂ + V₂V₃ + V₁V₃)

Unequal Time and Speed Calculations

• A particle moves with speed V₁ for half the time, and for the remaining half, it moves with speed V₂ then V₃, the average speed can be calculated with the formula: average speed = (2V₁(V₂ + V₃)) / (2*V₁ + V₂ + V₃)

Notes on Average Speed

• For straight-line motion without turning, the average speed is equal to the average velocity
• Otherwise, average speed is always greater than average velocity
• Instantaneous speed is always equal to the magnitude of instantaneous velocity
• Instantaneous velocity is always tangential to the trajectory of the particle

Acceleration

• Acceleration is defined as the rate of change of velocity with respect to time
• Acceleration is a vector quantity
• Acceleration is frame of reference dependent
• The S.I. unit of acceleration is m/s²

Acceleration Formulae

• If speed increases, there is acceleration
• If speed decreases, there is retardation or deceleration
• Average acceleration = (Vf - Vi) / (tf - ti), where Vf and Vi are the final and initial velocities, and tf and ti are the final and initial times
• Instantaneous acceleration is the derivative of velocity with respect to time, a = dv/dt