Kinematics of Particles

Position, velocity and acceleration

A particle is a body whose size does not matter to the question, such as a vehicle on a long track or a part in flight. Pick an origin and a positive direction first. Velocity is the rate of change of position and acceleration is the rate of change of velocity, so each is the slope of the one before it and each is the area under the one after it.

Eliminating time gives a third relation, a·ds = v·dv. Use it when acceleration is given as a function of position, or when a problem gives speeds and distances but no times.

Constant acceleration

When a is constant, integrating once and twice gives the three familiar equations: v = v₀ + a·t, s = s₀ + v₀·t + ½·a·t², and v² = v₀² + 2·a·(s − s₀). They only hold while a is constant. A motion with several phases, such as speed up, cruise and slow down, is solved one phase at a time, with the end of each phase as the start of the next.

Curvilinear motion

In two or three dimensions, velocity and acceleration are vectors. Rectangular components suit projectiles: horizontal motion has zero acceleration and vertical motion has a = −g, and the two share only the time. Normal and tangential (n–t) components suit motion along a known path: the tangential acceleration aₜ = dv/dt changes the speed, and the normal acceleration aₙ = v²/ρ, toward the centre of curvature, changes the direction.

Kinematics feeds the rest of dynamics. Once you know the acceleration, Newton’s second law gives the forces; the work–energy principle connects speed to position without finding the acceleration at all.

Key formulas

Worked examples

Common mistakes

• Using the constant-acceleration equations when a varies. Check that a is constant over the interval. If a depends on t, s or v, integrate v = ds/dt and a = dv/dt directly.
• Mixing sign conventions within one problem. Choose a positive direction at the start and give every s, v and a a sign. Deceleration is a negative a, not a positive one.
• Treating total distance as displacement. If a particle reverses direction, find where v = 0 and add the distances on each side of that point.
• Carrying a horizontal acceleration in projectile problems. With air resistance neglected, horizontal velocity is constant. Only the vertical motion has −g.
• Forgetting the normal acceleration on a curve. Even at constant speed, a particle on a curved path accelerates toward the centre at v²/ρ.

In industry

Frequently asked questions

Keep learning

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See also

• Engineering dynamics lessons
• Free body diagrams lesson
• Takt time calculator
• Engineering unit converter

Sources

1. R. C. Hibbeler, Engineering Mechanics: Dynamics, Pearson
2. F. P. Beer, E. R. Johnston, P. J. Cornwell, B. P. Self, Vector Mechanics for Engineers: Dynamics, McGraw Hill
3. J. L. Meriam, L. G. Kraige, J. N. Bolton, Engineering Mechanics: Dynamics, Wiley