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: The total length of the actual path traveled

• Scalar quantity (magnitude only)
• Always positive
• Path-dependent

• Displacement: The straight-line vector from initial to final position

• Vector quantity (magnitude and direction)
• Direction is from initial to final position
• Path-independent
• Can be smaller than the distance traveled

Example: When walking 3 km along a winding mountain trail that ends just 1 km east of your starting point, your distance traveled is 3 km, but your displacement is only 1 km east.

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.

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.

When speed increases linearly with time, acceleration is constant and equals the slope of the speed-time graph:

• Mathematically: a = Δv/Δt
• For the given values: a = (6 m/s - 0 m/s)/(3 s - 0 s) = 2 m/s²

Key insights:

• The slope of a speed-time graph represents acceleration
• Constant acceleration produces a straight line on a speed-time graph
• The steeper the slope, the greater the acceleration

Example: If a train's speed increases from 10 m/s to 25 m/s in 5 seconds, its acceleration is (25 m/s - 10 m/s)/5 s = 3 m/s².

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

The principle of superposition of motions (or principle of independence of motions) allows us to analyze 2D motion by separating it into independent x and y components.

Key aspects:

• Motion along each coordinate axis proceeds independently of motion along the other axis
• The x-component of motion depends only on the x-component of initial velocity and x-component of acceleration
• The y-component of motion depends only on the y-component of initial velocity and y-component of acceleration

Practical implications:

1. Complex 2D problems can be simplified into two 1D problems
2. Equations for 1D kinematics apply separately to each component:
   • vx = ux + axt and vy = uy + ayt
   • x = uxt + ½axt² and y = uyt + ½ayt²
3. Projectile motion analysis becomes straightforward by treating horizontal and vertical motions separately
4. Vector addition of the components fully reconstructs the complete 2D motion

This principle is why the trajectory of a projectile is parabolic - horizontal motion proceeds at constant velocity while vertical motion undergoes constant acceleration.

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.

This relationship exists because of the symmetry of the parabolic trajectory:

Time to maximum height: $t_{max} = \frac{u\sin θ}{g}$ (when vertical velocity becomes zero)

Total time of flight: $T = \frac{2u\sin θ}{g}$

Clearly, $t_{max} = \frac{T}{2}$

This occurs because:

• The vertical motion is symmetrical about the highest point
• Gravitational acceleration is constant
• The initial upward velocity equals the final downward velocity (at the same height)
• No air resistance is considered in the model

Acceleration transformation between frames moving with uniform relative velocity:

$\vec{a}_{P,S} = \vec{a}_{P,S'}$

Where:

• $\vec{a}_{P,S}$ is acceleration of particle P measured in frame S
• $\vec{a}_{P,S'}$ is acceleration of particle P measured in frame S'

This demonstrates the principle of Galilean invariance of acceleration:

• In non-relativistic mechanics, acceleration is the same in all inertial reference frames
• This is fundamental to Newton's laws, which take the same form in all inertial frames
• It means that while position and velocity measurements depend on the reference frame, acceleration measurements do not (when frames move with constant velocity)
• This invariance breaks down at very high speeds approaching the speed of light

This principle is crucial for understanding that the physical laws governing motion are the same for all observers in inertial reference frames, forming a cornerstone of classical mechanics.