2.2 Force and Newton's Second Law

Newton's Second Law

The net force on an object equals the rate of change of its momentum: ΣF = dp/dt. When mass is constant, this reduces to ΣF = ma. Because force and acceleration are vectors, this law can be applied independently along each coordinate axis (x, y) to solve multidimensional problems.

Variable Mass and Calculus Form

Since F = dp/dt = d(mv)/dt, if mass changes over time (like a rocket burning fuel), you must use the product rule: F = m(dv/dt) + v(dm/dt). This full form is essential in AP Physics C, distinguishing it from the simpler algebra-based F = ma used when mass is constant.

Free-Body Diagrams and Force Analysis

To apply Newton's Second Law, first draw a free-body diagram isolating the object and showing all forces acting on it. Sum the forces in each direction (often resolving vectors into x and y components), set the sum equal to ma in that direction, and solve. Acceleration can be found via integration or differentiation of position/velocity functions.

Key terms

Newton's Second Law (general form)

ΣF = dp/dt — the net force equals the rate of change of momentum.

Newton's Second Law (constant mass)

ΣF = ma — valid when mass does not change over time.

Variable mass force equation

F = m(dv/dt) + v(dm/dt), derived using the product rule on F = d(mv)/dt.

Free-body diagram

A diagram isolating an object and showing all external forces acting on it, used to apply Newton's Second Law.

Vector nature of F = ma

Force and acceleration are vectors, so Newton's Second Law can be applied separately along each axis (x and y components).

Try an AP-style question

A force F(t) = 6t N acts on a 2 kg object at rest. What is its speed at t = 2 s?

1. 24 m/s
2. 12 m/s
3. 3 m/s
4. 6 m/s