Part II: Dynamics · Topic 5 of 8 · 20 min read

Particle Kinematics

Describe motion geometrically before introducing the forces that produce it.

What you'll learn

Relate position, velocity and acceleration through differentiation and integration.
Solve problems with time-dependent acceleration or piecewise motion.
Resolve two-dimensional projectiles into independent x and y motion.
Use normal/tangential components to explain curved-path acceleration.

A velocity against time graph. A smooth curve starts high on the v axis and falls slowly at first, then faster, staying above the t axis. Lightly shade the area under the curve in pale blue. Draw a short dark red straight tangent arrow touching the curve near the middle. Labels: v on the vertical axis; t on the horizontal axis; slope = a(t) beside the tangent. Inside the shaded area: Area below v(t) = displacement

Understand the concepts

Kinematics describes motion

Position $\mathbf r(t)$ locates the particle; velocity is its time derivative and acceleration is the derivative of velocity. A negative acceleration component does not always mean that the speed decreases.

One coordinate can hide the picture

The integral constants are fixed by initial conditions. Plotting $s(t)$ , $v(t)$ and helps establish when a particle reverses direction or momentarily stops.

Assumptions and sign conventions

Particle model (size and rotation neglected).
In projectiles, air resistance is neglected unless specified.
Constant- $g$ near Earth; stationary Cartesian axes unless otherwise specified.
Normal/tangential unit vectors follow the instantaneous trajectory.

Equations that matter

\mathbf v=\frac{d\mathbf r}{dt},\qquad \mathbf a=\frac{d\mathbf v}{dt}

Step-by-step method

Sketch the trajectory and choose signed coordinates.
List initial conditions and decide whether acceleration is constant, a function of time, or a function of position.
Differentiate/integrate with consistent units and apply initial conditions immediately.
For projectile motion, use one time coordinate in both axes and remember $v_y=0$ only at the peak.
For curved motion, determine instantaneous speed and curvature before forming tangential/normal acceleration.

Worked example: Time-dependent braking

A testing capsule enters a resistive medium at $12.0\,\mathrm{m/s}$ along a straight axis. It experiences acceleration $a(t)=-4t\,\mathrm{m/s^2}$ ( in seconds). Find the time and distance travelled before it first stops.

Common mistakes

Using constant-acceleration formulas for $a(t)$ that changes with time.
Setting acceleration, rather than vertical velocity, to zero at a projectile apex.
Confusing scalar speed with signed velocity.
Treating normal acceleration as pointing outward from a curve.

Check yourself

Can an object accelerate while maintaining constant speed?
At a projectile apex, which velocity and acceleration components are zero?
When does displacement differ from total distance travelled?

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Related: Rigid-Body Kinematics

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Particle Kinematics

What you'll learn

Understand the concepts

Kinematics describes motion

One coordinate can hide the picture

Assumptions and sign conventions

Equations that matter

Step-by-step method

Worked example: Time-dependent braking

Common mistakes

Check yourself

Separate projectile axes

Curvature matters even at constant speed

Relative motion is vector subtraction

Step 1: Integrate acceleration

Step 2: Find the first stopping time

Step 3: Integrate velocity

Step 4: Evaluate at stopping time