Questions: Rectilinear Motion of Particles

Question 1

A particle's acceleration is given as a function of position, a = f(x). You want to find velocity as a function of position without explicitly solving for x(t). Which relationship is most useful?

• A. v = dx/dt, integrated directly
• B. a = dv/dt, integrated with respect to time
• C. a = v dv/dx, which allows separating v and x
• D. v² = v₀² + 2a(x − x₀), valid for all accelerations

Answer: (C)

When acceleration is a function of position, the chain rule identity a = v dv/dx lets you write f(x) = v dv/dx and separate variables: f(x)dx = v dv, integrating both sides to get v as a function of x directly. The constant-acceleration equation v² = v₀² + 2a(x − x₀) only applies when a is constant. Integrating a = dv/dt with respect to time requires knowing a as a function of t, not x. The a = v dv/dx form is precisely the tool for position-dependent acceleration.

Question 2

A car brakes uniformly from v₀ = 30 m/s to rest. The braking distance is 45 m. Which kinematic equation most directly gives the deceleration, and why?

• A. v = v₀ + at, because it directly relates velocity, acceleration, and time
• B. x = x₀ + v₀t + ½at², because it includes displacement
• C. v² = v₀² + 2a(x − x₀), because it relates velocity, acceleration, and displacement without needing time
• D. x = x₀ + ½(v₀ + v)t, because it uses the average velocity

Answer: (C)

Here the knowns are v₀ = 30 m/s, v = 0 m/s, and Δx = 45 m; the unknown is a. The equation v² = v₀² + 2aΔx involves exactly these four quantities and no time — substituting gives 0 = 900 + 90a, so a = −10 m/s². The other equations all involve time (t), which is not given and would require a second step to eliminate. Each of the four constant-acceleration equations omits one of the five kinematic quantities, so choosing the right equation means identifying which quantity is absent.

Question 3

The four constant-acceleration kinematic equations are derived by integrating a = constant twice, and each requires initial conditions to determine the constants of integration.

• T. True
• F. False

Answer: True

Starting from a = constant: integrating once gives v = v₀ + at (v₀ is the integration constant, the initial velocity). Integrating again gives x = x₀ + v₀t + ½at² (x₀ is the second integration constant, the initial position). The other two equations (v² = v₀² + 2aΔx and x = ½(v₀+v)t) are algebraic combinations of these two. This derivation shows exactly where the equations come from and why they are limited to constant acceleration — the integration is valid only when a is not a function of t.

Question 4

Kinematics requires knowledge of the forces acting on a particle in order to determine its position and velocity over time.

• T. True
• F. False

Answer: False

Kinematics is entirely force-free — it describes motion (position, velocity, acceleration as functions of time) without any reference to what causes the motion. Forces appear in kinetics, not kinematics. In a typical dynamics problem, the two are combined: kinetics (Newton's second law ΣF = ma) provides the acceleration, and kinematics then predicts position and velocity from that acceleration. But the kinematic equations themselves — including the four constant-acceleration equations — make no mention of mass, force, or any physical cause. This separation is conceptually important and practically useful.

Question 5

What is the conceptual difference between kinematics and kinetics, and why is it useful to keep them distinct when solving motion problems?

Model answer: Kinematics describes *how* a particle moves — it defines the relationships between position, velocity, acceleration, and time through calculus, without asking why. Kinetics explains *why* the motion occurs — it applies Newton's second law to relate forces and mass to acceleration. In problem-solving, keeping them distinct creates a clean two-step process: (1) use force analysis (kinetics) to determine acceleration from the net force and mass, then (2) use kinematic equations to find position and velocity from that acceleration. Conflating them leads to circular reasoning or skipping force analysis entirely.

The distinction also clarifies what information is needed at each step. The kinematic equations work for *any* motion with a given acceleration profile — they don't care what caused the acceleration. The kinetics step is where physical understanding (friction coefficients, spring constants, gravity) enters. Once you have a(t), a(x), or constant a, kinematics takes over and the physics is done. This modularity is why engineering dynamics courses teach kinematics first, then kinetics — the tools are separate and each must be mastered independently.