Kinematics: Constant Acceleration and Calculus in One-Dimensional Motion

One-Dimensional Motion and the Foundations of Kinematics

Problem setup (one-dimensional motion): an object moves along a horizontal line (x-axis). If you know the position at time t1, x1, and you wait until a later time t2, the position is x2. The question is to solve for x2 given x1, t1, and t2 (e.g., t2 = t1 + 3.5 s).
Algebraic formulation of the simple (constant-velocity) case: the question
- In words: where will the object be at time t2 if it starts at x1 at t1 with velocity v?
- In equation form (constant velocity):
  x2 = x1 + v\,(t2 - t1)
Commutativity reminder: the order of terms in addition or multiplication does not affect the result (commutative property).
Important caveat: the equation x2 = x1 + v(t2 − t1) is valid only if the speed is constant. If the speed changes with time, this form cannot be used directly; you would need the average speed over the interval or, more generally, calculus when acceleration is present.
Initial example tying to real numbers (illustrative): if the initial position is x1 = 3.1 m and the observation starts at t1 = 11.5 s, knowing the speed and the elapsed time allows you to compute x2, provided the speed is constant over the interval.
The historical motivation for introducing acceleration:
- When speed is not constant, you need a new parameter to describe how the speed changes. That parameter is acceleration, defined as the rate of change of velocity.
- Acceleration has units of m/s^2 and can be instantaneous or average, just as velocity can be instantaneous or average.
Units and meaning of acceleration:
- Numerator: velocity changes, with units of m/s.
- Denominator: time, with units of s.
- Therefore, the units of acceleration are m/s^2, which can be thought of as how many meters per second the velocity changes each second (the rate of change of velocity).
- Physically: positive acceleration means velocity increasing; negative acceleration means velocity decreasing.
Constant vs non-constant acceleration:
- The simple kinematic equations below are derived assuming acceleration is constant.
- If acceleration varies with time, you must use the general integral form or a detailed time-dependent model a(t).
The historical context of science leading to calculus:
- Early ideas about motion and distance were based on daily experience (e.g., walking to a watering hole at a roughly constant pace).
- Galileo (around 1600) performed careful experiments and began formulating laws of motion, including relationships for pendulums, and emphasized controlled experiments.
- Isaac Newton (late 17th century) applied mathematics to describe motion with precise equations, leading to the birth of calculus to handle changing quantities (velocity, acceleration).
- The invention of calculus allowed precise expressions for motion with changing velocity (not just constant speed).
Infinitesimals and the Calculus foundation (conceptual view):
- dx and dt are infinitesimally small changes in position and time, respectively.
- The ratio dx/dt defines instantaneous velocity; dx and dt individually have no fixed numerical value, but their ratio is a real number.
- One can think of an infinitesimal dt as a very small time interval during which velocity is effectively constant; summing many such tiny changes over a finite interval yields finite displacement.
- Physically, an infinitesimal instantaneous moment occurs, for example, at the apex of a projectile’s trajectory, where the duration of that moment is infinitesimal.
From infinitesimals to integrals:
- The change in position over a time interval is obtained by summing all the tiny displacements:
  dx = v\,dt\quad\Rightarrow\quad x2 - x1 = \int{t1}^{t_2} v(t)\,dt.
- The integral arises as a smooth summation of infinitely many infinitesimal contributions in time.
- When the speed is constant, v can be pulled out of the integral:
  x2 - x1 = v\int{t1}^{t2} dt = v\,(t2 - t_1)
- The constant of integration appears when you perform an indefinite integral; for a problem with a known starting point, that constant is fixed by the boundary condition x(t1)=x1.
Boundary conditions and initial data:
- To solve a motion problem, you need boundary conditions such as the initial position x1 and initial velocity v1 (or v0, vinitial, vnaught, etc.).
- With x(0) = x0 and v(0) = v0 (taking t1 = 0 for convenience), you can express x and v as functions of time once acceleration a(t) is specified.
Constant acceleration: the two fundamental equations (and a third derived form)
- Velocity as a function of time (constant a):
  v(t) = v_0 + a\,t.
- Position as a function of time (constant a):
  x(t) = x0 + v0\,t + \tfrac{1}{2}\,a\,t^2.
- These assume acceleration a is constant; otherwise, you must integrate a(t) appropriately.
- A useful acceleration relation that eliminates time (also valid for constant a):
  v^2 = v0^2 + 2\,a\,(x - x0).
- Relationship among the three equations arises by integrating the velocity expression and substituting into the position integral, or by eliminating t between the v(t) and x(t) expressions.
Notation caveats and practical tips:
- Different courses and textbooks use v1, v0, vinitial, vnaught, etc.; they all denote the initial velocity at t = 0 for that problem.
- When solving problems, you may set t1 = 0 to simplify and denote the later time simply as t.
- The constant a must be identified from given data before using the constant-acceleration equations; otherwise, those equations are not applicable.
Example problem (Chapter 2, Problem 30): a block on a slope with constant acceleration
- Given: constant acceleration, one-dimensional motion; initial velocity v0 = 0 (released from rest); initial position x0 = 0 (choose a convenient origin).
- Observations: after traveling Δx = 6.8 m, the speed is v = 3.8 m/s.
- They ask for the speed when the block has traveled Δx = 3.4 m (halfway down).
- Step 1: Use the v^2 relation to find the acceleration a from the first given data:
  v^2 = v0^2 + 2\,a\,(x - x0)\quad\Rightarrow\quad (3.8)^2 = (0)^2 + 2\,a\,(6.8).
- Solve for a:
  a = \frac{(3.8)^2}{2\cdot 6.8} \approx 1.062\ \text{m/s}^2.
- Step 2: Use the same v^2 relation to find v at x = 3.4 m:
  v^2 = v0^2 + 2\,a\,(x - x0) = 0 + 2\,a\,(3.4).
- Compute: v^2 \approx 2\cdot 1.062\cdot 3.4 \approx 7.214\quad\Rightarrow\quad v \approx \sqrt{7.214} \approx 2.69\ \text{m/s}.
- If one uses time explicitly: with v = v0 + a t and x = x0 + v0 t + (1/2) a t^2, solve for t when x = 3.4 m (with v0 = 0): x = \tfrac{1}{2} a t^2 \Rightarrow t = \sqrt{\tfrac{2x}{a}}\Rightarrow t \approx \sqrt{\tfrac{2\cdot 3.4}{1.062}} \approx 2.53\ \text{s}, then v = v0 + a t \approx 1.062\cdot 2.53 \approx 2.69\ \text{m/s}.
- Comparison insight: at half the distance, the speed is not half of the final speed; rather, since v^2 ∝ x for constant a, halving distance gives v ≈ v_final / \sqrt{2} ≈ 3.8 / \sqrt{2} ≈ 2.69\ \text{m/s}.
Summary of key takeaways
- x2 − x1 = ∫ from t1 to t2 of v(t) dt; for constant velocity this reduces to x2 = x1 + v (t2 − t1).
- Acceleration a is the constant rate of change of velocity; velocity changes as v(t) = v0 + a t; position changes as x(t) = x0 + v0 t + (1/2) a t^2 when a is constant.
- The v^2 relation, v^2 = v0^2 + 2 a (x − x0), provides a direct link between velocity and displacement without explicit time dependence.
- Boundary/boundary conditions (initial position x0 and initial velocity v0) set the integration constants and give complete solutions.
- The equations above are foundational for classical mechanics and were central to Newton’s development of calculus and the mathematical description of motion; they also underpin practical problems such as gravity (g ≈ 9.8 m/s^2 on Earth).
Practical context and limits
- These constant-acceleration equations are especially applicable to problems like gravity near the Earth, projectiles, and other systems where a is approximately constant over the interval of interest.
- When acceleration is not constant, one must revert to the general integral forms, e.g., x2 − x1 = ∫_{t1}^{t2} v(t) dt with v(t) = v0 + ∫ a(t) dt, or use the velocity-integral form with a(t).
Historical and real-world relevance (brief recap):
- Galileo’s empirical work laid the groundwork for quantifying motion; Newton synthesized the observations with calculus to derive the governing equations.
- The methods enable predictions about motion ranging from a dropped book to space missions (e.g., launch and cruise phases involve substantial accelerations where propulsion and gravity interplay).
- The same mathematical framework underpins many technologies and scientific advances, illustrating how precise equations translate into real-world design and exploration.
Final note on problem-solving approach
- Always identify whether acceleration is constant before choosing the appropriate set of equations.
- Use boundary conditions to fix constants of integration (e.g., x0, v0 at t = t1).
- When solving for a variable at a given position, you can use the v^2 relation to avoid explicitly solving for time, or derive t and substitute back as needed.
- Recognize multiple notations (v1, v0, vinitial, vnaught) and adapt to the symbol in the problem statement.
Quick reference equations (constants and common forms):
- Displacement with constant velocity:
  x2 - x1 = v\,(t2 - t1).
- Velocity with constant acceleration:
  v(t) = v_0 + a\,t.
- Position with constant acceleration:
  x(t) = x0 + v0\,t + \tfrac{1}{2}\,a\,t^2.
- Velocity-displacement relation (no explicit time):
  v^2 = v0^2 + 2\,a\,(x - x0).
- Displacement via integral (general case):
  x2 - x1 = \int{t1}^{t_2} v(t)\,dt.