Rest and Motion : Kinematics

• A frame of reference is a coordinate system used to specify the position of objects and measure their motion

• Minimum components required:

1. A set of three mutually perpendicular axes (X-Y-Z) to locate positions in space
2. A specified origin point (0,0,0) from which all positions are measured
3. A clock or time-keeping system to measure when events occur
4. A rigid body to which the coordinate system is attached

• A complete reference frame allows:

• Position to be specified using coordinates (x,y,z)
• Motion to be described as change in coordinates with time
• Velocity and acceleration to be measured relative to this frame

• Example: When analyzing motion in a train, we might define a frame attached to the train with:

• Origin at a specific point in the train
• X-axis along the train's length
• Y-axis across the width
• Z-axis vertically
• Time measured by a clock in the train

Note: The choice of frame is arbitrary and depends on what makes the analysis most convenient

Distance calculation: • Sum the lengths of all path segments along the actual trajectory • Mathematically: $s = \int_{path} |d\vec{r}|$ • For discrete segments: $s = \sum |Δ\vec{r}_i|$

Displacement calculation: • Vector connecting initial point A to final point B • Mathematically: $\vec{D} = \vec{r}_B - \vec{r}_A$ • Magnitude equals the straight-line distance between A and B • Direction points from A to B

Note: Distance ≥ Displacement (magnitude), with equality only when the path is a straight line.

• Average speed and average run rate share the same mathematical principle: a total quantity divided by time
• Average speed = total distance traveled / total time elapsed
• Similarly, average run rate = total runs scored / total overs played
• Both are aggregate measures that smooth out variations in instantaneous rates
• Both provide overall performance metrics while hiding moment-to-moment fluctuations
• Example: A vehicle traveling 300 km in 4 hours has an average speed of 75 km/h, regardless of whether it maintained constant speed or varied between faster and slower periods

Note: The instantaneous rate at any specific moment may differ significantly from the average rate over the entire interval.

On a distance-time curve:

• Average speed between two times ($t$ and $t + \Delta t$) corresponds to the slope of the chord connecting the two corresponding points on the curve
• Instantaneous speed at time $t$ corresponds to the slope of the tangent to the curve at that point

Mathematical relationship:

• As $\Delta t$ approaches zero, the chord approaches the tangent
• The slope of the chord ($\frac{\Delta s}{\Delta t}$) approaches the slope of the tangent ($\frac{ds}{dt}$)

This geometric interpretation shows why: $v = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t} = \frac{ds}{dt}$

Note: The tangent represents the "best linear approximation" of the curve at that point, showing the instantaneous rate of change.

For an object with linearly increasing speed (constant acceleration):

1. Speed as a function of time:

   • v(t) = at
   • v(t) = 2t m/s (where a = 2 m/s² is the acceleration)

2. Position as a function of time:

   • x(t) = x₀ + (1/2)at²
   • x(t) = (1/2)(2 m/s²)t² = t² m

This represents uniformly accelerated motion where:

• The object starts from rest (v₀ = 0)
• The acceleration is constant (a = 2 m/s²)
• The displacement after 3 seconds equals 9 meters
• The average speed equals half the final speed: vₐᵥₑ = 3 m/s

Note: These equations apply to straight-line motion with constant acceleration, such as objects in free fall (ignoring air resistance) or vehicles with steady throttle input.

When you know initial velocity (u), acceleration (a), and displacement (x), but need to find final velocity (v), you should use:

v² = u² + 2ax

This equation is ideal for this specific scenario because:

1. It directly relates the variables you know (u, a, x) to the one you need to find (v)
2. It doesn't require time (t), which you don't have
3. You can solve for v by taking the square root of both sides: v = √(u² + 2ax)

Example application: A car accelerates from 5 m/s at 2 m/s² over a distance of 100 m. Its final velocity would be: v = √(5² + 2×2×100) = √(25 + 400) = √425 ≈ 20.6 m/s

A projectile's motion consists of two independent components:

1. Horizontal motion:

   • Constant velocity (u·cosθ)
   • Zero acceleration (ax = 0)
   • Distance: x = (u·cosθ)t

2. Vertical motion:

   • Changing velocity (initial: u·sinθ)
   • Constant downward acceleration (ay = -g)
   • Height: y = (u·sinθ)t - (1/2)gt²

This independence of horizontal and vertical components is why projectile motion follows a parabolic path, with gravity only affecting the vertical component.

Example: A baseball thrown horizontally from a cliff moves at constant horizontal speed while simultaneously accelerating downward due to gravity.

Range = horizontal distance traveled during time of flight.

Range = horizontal velocity × time of flight $R = (u\cos θ) \times \frac{2u\sin θ}{g}$ $R = \frac{2u^2\sin θ\cos θ}{g}$

Using the identity $\sin 2θ = 2\sin θ\cos θ$: $R = \frac{u^2\sin 2θ}{g}$

To maximize range, $\sin 2θ$ must be maximum. Since $\sin 2θ$ has a maximum value of 1 when $2θ = 90°$, the range is maximum when $θ = 45°$.

Maximum range: $R_{max} = \frac{u^2}{g}$ (when $θ = 45°$)

To calculate the resultant velocity when swimming perpendicular to a river current:

1. Use vector addition of the two velocity components:

   • Swimmer's velocity relative to water (vS,W)
   • River's velocity relative to ground (vR,G)

2. Apply the Pythagorean theorem: $v_{resultant} = \sqrt{v_{S,W}^2 + v_{R,G}^2}$

3. Calculate the angle of motion relative to the current: $\theta = \tan^{-1}\left(\frac{v_{S,W}}{v_{R,G}}\right)$

Example: A swimmer moving at 4.0 km/h perpendicular to a 3.0 km/h current will have a resultant velocity of 5.0 km/h at an angle of 53.1° relative to the current direction.

The relationship between relative velocities in different reference frames follows these principles:

1. Vector addition law: $\vec{v}_{A,C} = \vec{v}_{A,B} + \vec{v}_{B,C}$ Where:

   • $\vec{v}_{A,B}$ is velocity of object A relative to reference frame B
   • $\vec{v}_{B,C}$ is velocity of reference frame B relative to reference frame C
   • $\vec{v}_{A,C}$ is velocity of object A relative to reference frame C

2. In flowing media (water/air):

   • A person's velocity relative to ground equals vector sum of their velocity relative to the fluid plus the fluid's velocity relative to ground
   • Direction is determined by the vector resultant, not simply the intended direction of travel

3. Practical applications:

   • Swimming across rivers
   • Aircraft navigation in wind
   • Boat navigation in currents