04 // Interactive Formula Compendium // PHYSICSF_net = m · a = dp / dt
Newton's Second Law of Motion & SUVAT Kinematics Equations Sheet
Newton's Second Law states that the net vector force acting on a body equals the time rate of change of its momentum, F_net = dp/dt = m·a (in Newtons, 1 N = 1 kg·m/s²).
Formula Reference
Main Formula
F_net = m · a = dp / dt
Rearranged Forms
v = v₀ + a·t (SUVAT Velocity-Time)
Δx = v₀·t + ½ a·t² (SUVAT Displacement-Time)
v² = v₀² + 2a·Δx (Time-Independent Torricelli Equation)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| F_net | Net External Force (F = m·a) | N |
| m | Inertial Mass (m) | kg |
| a | Linear Acceleration (a) | m/s² |
Inputs & Parameters
[kg]
[m/s²]
Calculated Result (Net External Force (F = m·a))
F_net =5,250N
Formula Notes & Background
- Define linear momentum p = m v; for constant mass m, differentiating with respect to time gives dp/dt = m(dv/dt) = m a.
- Integrate constant acceleration a = dv/dt over [0, t] to get v(t) = v₀ + at.
- Integrate v(t) = dx/dt over [0, t] to get Δx = v₀t + ½at², and eliminate t between the two equations to obtain v² = v₀² + 2aΔx.
Step-by-Step Examples
#1: Find the net propulsive force needed to accelerate a 1,500 kg electric vehicle at 3.5 m/s².
m = 1,500 kga = 3.5 m/s²
F_net = 1500 kg × 3.5 m/s²
→ F_net = 5,250 N (5.25 kN)
#2: How far does the vehicle travel while accelerating from rest (v₀ = 0) at 3.5 m/s² for t = 6.0 s?
v₀ = 0 m/sa = 3.5 m/s²t = 6.0 s
Δx = 0 + ½ × 3.5 × 6.0² = 1.75 × 36
→ Δx = 63.0 m (reaching final speed v = 21.0 m/s)
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Frequently Asked Questions
What is the difference between mass m and weight W?+
Mass m (in kg) is an intrinsic scalar measure of inertia everywhere in the universe, whereas weight W = m·g (in Newtons) is the local gravitational force acting on that mass.
When does F = ma fail and require the general form F = dp/dt?+
Variable-mass systems (such as rockets expelling exhaust via the Tsiolkovsky rocket equation) and relativistic particles approaching the speed of light require F = dp/dt.
Which SUVAT equation should you choose when time t is unknown?+
Use Torricelli's equation v² = v₀² + 2aΔx, which links initial speed, final speed, acceleration, and distance without time t.