Rest and Motion : Kinematics

The relationship between accelerations in two reference frames S and S' is:

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

Where:

• $\vec{a}_{P,S}$ is the acceleration of particle P relative to frame S
• $\vec{a}_{P,S'}$ is the acceleration of P relative to frame S'
• $\vec{a}_{S',S}$ is the acceleration of frame S' relative to frame S

A body has identical acceleration in both frames when: $\vec{a}_{S',S} = 0$

This occurs when:

• The frames move with uniform velocity relative to each other
• There is no relative acceleration between the frames
• The frames are in translational motion with constant velocity

This is a key principle of Newtonian relativity: the laws of mechanics are the same in all inertial reference frames (frames moving at constant velocity relative to each other).

• 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.

Instantaneous speed is defined as the limit of the average speed as the time interval approaches zero:

$v = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t} = \frac{ds}{dt}$

Where:

• $v$ is the instantaneous speed
• $\Delta s$ is the distance traveled in time $\Delta t$
• $\frac{ds}{dt}$ is the derivative of the distance function with respect to time

In physical terms, instantaneous speed represents the rate at which distance is changing at a specific moment in time, rather than over an interval.

Example: A car's speedometer shows instantaneous speed at each moment, not the average speed over a journey.

1. Total distance traveled: $s = \int_{t_1}^{t_2} v(t)\,dt$

2. Average speed: $v_{avg} = \frac{s}{t_2 - t_1} = \frac{\int_{t_1}^{t_2} v(t)\,dt}{t_2 - t_1}$

Relationship:

• The average speed equals the total distance divided by the total time
• Geometrically, average speed equals the height of a rectangle with width (t₂-t₁) that has the same area as under the speed-time curve

Example: If a car travels 120 km in 2 hours with varying speed, its average speed is 60 km/h regardless of how its actual speed varied during the journey.

Note: The average speed is not necessarily equal to the arithmetic mean of speeds at different times, especially when acceleration isn't constant.

The three fundamental kinematic equations for constant acceleration are:

1. v = u + at

   • Relates final velocity (v) to initial velocity (u) through acceleration (a) and time (t)

2. x = ut + ½at²

   • Gives position (x) in terms of initial velocity (u), time (t), and acceleration (a)

3. v² = u² + 2ax

   • Relates velocity squared to position without requiring time as a variable

These equations form the mathematical foundation of linear motion with constant acceleration and allow us to solve for any variable when given three of the five kinematic variables (u, v, a, t, x).

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.

The range R of a projectile launched from and landing at the same horizontal level is:

$R = \frac{u^2 \sin(2\theta)}{g}$

Key insights:

• Range depends on the square of the initial velocity (u²)
• Range is proportional to sin(2θ), reaching maximum at θ = 45°
• For any range less than maximum, there are two complementary angles that yield the same range
• Gravitational acceleration g is inversely proportional to range

For projectiles launched and landing at different heights:

• The asymmetry alters the optimal angle from 45°
• Launching from higher to lower ground increases range
• Launching from lower to higher ground decreases range

Example: A golf ball struck with initial velocity 40 m/s at 30° angle will travel approximately: R = (40²·sin(60°))/9.8 = 1600·0.866/9.8 ≈ 141.6 meters

Range: $R = \frac{u^2\sin 2θ}{g} = \frac{(12.0)^2 \times \sin(2 \times 45°)}{10.0} = \frac{144 \times 1}{10.0} = 14.4$ m

Maximum height: $H = \frac{u^2\sin^2 θ}{2g} = \frac{(12.0)^2 \times \sin^2(45°)}{2 \times 10.0} = \frac{144 \times 0.5}{20.0} = 3.6$ m

Time of flight: $T = \frac{2u\sin θ}{g} = \frac{2 \times 12.0 \times \sin(45°)}{10.0} = \frac{2 \times 12.0 \times 0.707}{10.0} ≈ 1.7$ s

If a particle P is observed from frames S and S', with S' moving relative to S, then:

Velocity relation: $\vec{v}_{P,S} = \vec{v}_{P,S'} + \vec{v}_{S',S}$

Rearranged: $\vec{v}_{P,S'} = \vec{v}_{P,S} - \vec{v}_{S',S}$

This means:

• The velocity of a particle with respect to frame S equals its velocity with respect to frame S' plus the velocity of frame S' with respect to S
• The velocity of particle relative to body 2 equals velocity of particle relative to body 1 minus velocity of body 2 relative to body 1

Practical applications:

• Calculating a swimmer's velocity relative to shore when swimming in moving water
• Determining a bullet's velocity relative to ground when fired from a moving vehicle
• Finding an airplane's velocity relative to the ground when flying in wind