Kinematics and Dynamics: Newton's Laws

Kinematics describes HOW an object moves, without worrying about the cause of the motion. Three quantities are enough: position (where the object is), velocity (how fast and in which direction its position changes), and acceleration (how fast the velocity itself changes). For uniform rectilinear motion, velocity is constant and position evolves linearly with time. For uniformly accelerated motion (free fall, for example), acceleration is constant and velocity evolves linearly with time.

Circular motion deserves particular attention for the IMAT: even at constant speed (magnitude), an object going around a circle is accelerating, because its direction keeps changing. This acceleration, directed toward the center of the circle, is called centripetal acceleration.

Dynamics explains WHY an object moves this way — that's the subject of Newton's three laws. The first (the law of inertia) states that an object stays at rest or in uniform rectilinear motion as long as no net force acts on it. The second relates force and acceleration: $F = m \cdot a$, the net force produces an acceleration proportional to the force and inversely proportional to the mass. The third (action-reaction) states that if a body A exerts a force on a body B, B exerts on A a force of equal magnitude, same direction, but opposite sense.

An essential point for the IMAT: it is never ONE isolated force that determines motion, but the resultant (vector sum) of ALL applied forces. A motionless object is not necessarily subject to no force at all — it may be subject to several forces that exactly cancel out (equilibrium).

Illustration: projectile trajectory

Calculation: time of flight, maximum height and range

A projectile is launched at $v_0 = 20\ \text{m/s}$ at $\theta = 30°$ above the horizontal, $g = 9.8\ \text{m/s}^2$. First resolve the initial velocity: $v_{0x} = v_0\cos\theta = 20 \times \cos 30° \approx 17.3\ \text{m/s}$ and $v_{0y} = v_0\sin\theta = 20 \times 0.5 = 10\ \text{m/s}$.

Time to reach the peak (vertical velocity zero): $t_{up} = \dfrac{v_{0y}}{g} = \dfrac{10}{9.8} \approx 1.02\ \text{s}$, so total flight time $t = 2t_{up} \approx 2.04\ \text{s}$ (by symmetry, landing at the same altitude). Maximum height: $h = \dfrac{v_{0y}^2}{2g} = \dfrac{100}{19.6} \approx 5.10\ \text{m}$. Range: $R = v_{0x} \times t \approx 17.3 \times 2.04 \approx 35.3\ \text{m}$.

A classic IMAT trap: forgetting that $v_{0y}$ is nonzero at launch (only the vertical velocity at the peak is zero), and confusing the horizontal acceleration ($a_x = 0$, since $\sum F_x = 0$ neglecting air) with the constant vertical acceleration $a_y = -g$ — this independent treatment on each axis follows directly from Newton's second law, $\sum \vec{F} = m\vec{a}$, applied separately per axis.

• Position, velocity, acceleration — three descriptions linked by successive derivatives
• Circular motion: constant speed magnitude but non-zero centripetal acceleration
• Newton's 1st law: inertia, no net force = no change in motion
• 2nd law: $F = m \cdot a$, the resultant of the forces determines the acceleration
• 3rd law: forces always come in action-reaction pairs, acting on TWO different bodies

• $v = \dfrac{\Delta x}{\Delta t}$ (average velocity)
• $a = \dfrac{\Delta v}{\Delta t}$ (average acceleration)
• $x = x_0 + v_0 t + \dfrac{1}{2}at^2$ (position, constant acceleration)
• $v^2 = v_0^2 + 2a\Delta x$ (velocity-displacement relation without time)
• $F = m \cdot a$ (Newton's 2nd law)
• $a_c = \dfrac{v^2}{r}$ (centripetal acceleration)

Think of the 1st law as the "law of laziness": an object never changes its own motion by itself, a force is needed to make it. The 3rd law is a common trap: when you push on a wall, the wall pushes back with exactly the same force — these are not two forces cancelling each other out (they act on two different objects, you and the wall), so nothing stops each of them from having its own effect.

• Confusing velocity and acceleration: a constant speed magnitude (circular motion) can still come with a non-zero acceleration
• Forgetting that a zero resultant doesn't mean no forces, but a balance of forces
• Applying $F = m \cdot a$ with a single force instead of THE resultant of all forces
• Confusing the two forces of an action-reaction pair with two forces cancelling out on the same object