# Newton's laws of motion

Newton's first law of motion states that the velocity of an object remains constant unless a force acts on it. The law essentially describes the property of **inertia**.

This means that

- if an object is
**stationary**$$(v=0\text{ m/s})$$, a force is needed to make it move. - if an object is moving, a force is needed to stop it or to change its speed or direction.

The law can be represented by: $$$\vecphy{F}=0 \Rightarrow \vecphy{v}=\text{constant}$$$ $$\vecphy{F}=$$ force; $$\vecphy{v}=$$ velocity.

Newton's first law sometimes goes ** against our intuition**. If you push an object, it will not continue moving forever. The law may surprise us because we tend to forget about friction.

Newton's second law of motion says that the acceleration $$\vecphy{a}$$ of an object is inversely proportional to its mass $$m$$ and directly proportional to the force $$\vecphy{F}$$ acting on it: $$$\vecphy{F}=m\vecphy{a}$$$

To achieve a certain acceleration, a big truck requires a much bigger force than what a small motorbike requires.

Newton's second law implies Newton's first law. Newton's second law says that if the force is zero, the acceleration is also zero. We know that if the acceleration is zero, the velocity of the object is constant:$$$\vecphy{F}=0 \Rightarrow \vecphy{a}=0 \Rightarrow \vecphy{v}=\text{constant}$$$

**Newton's second law of motion** is usually stated as $$\vecphy{F}=m\vecphy{a}$$, but it can also be stated in terms of the momentum $$\vecphy{p}$$.

The rate of change of **momentum** of an object is directly proportional to the force acting on the object: $$$\frac{d\vecphy{p}}{dt} = \vecphy{F}$$$

The equivalence of the two versions of the law can be shown by starting with $$$\vecphy{p}=m\vecphy{v}$$$ where $$m$$ is assumed to be constant.

Acceleration is equal to the change in velocity with respect to time: $$$\vecphy{a}= \frac{d\vecphy{v}}{dt}$$$ According the definition of momentum, this suggests that $$$\frac{d\vecphy{p}}{dt}=m\frac{d\vecphy{v}}{dt}=m\vecphy{a}$$$ This leads to the conclusion that $$$\vecphy{F}=m\vecphy{a}=\frac{d\vecphy{p}}{dt} $$$

$$\vecphy{F}=$$force acting on an object; $$m=$$ mass of the object; $$\vecphy{a}=$$ acceleration of the object as a result of the force acting on it; $$\vecphy{p}=$$ momentum; $$\vecphy{v}=$$velocity, $$t=$$time.

** Force ** and ** impulse ** are closely related concepts. Impulse is normally defined as the change in momentum, but it can also be defined in terms of force.

An impulse of $$10 \text{ kg m/s}$$ could be the result of a force of $$1 \text{ N}$$ applied over $$10 \text{ s}$$ or of a force of $$10 \text{ N}$$ applied over $$1 \text{ s}$$.

If mass and force are constant during the period of impact, the impulse is also directly proportional to the force and the change in velocity: $$$\vecphy{J}=\Delta \vecphy{p}=\vecphy{F}t$$$ This can be shown as follows: $$$ \begin{align*} \vecphy{J}&=\Delta\vecphy{p}&\quad\quad \vecphy{F}&=\frac{d\vecphy{p}}{dt}&\\ &=\vecphy{p}_1-\vecphy{p}_0&\quad\quad &=\frac{\Delta\vecphy{p}}{t}&\\ &=m(\vecphy{v}_1-\vecphy{v}_0)\\ &=m\Delta\vecphy{v}\\ \Rightarrow\vecphy{J}&=\Delta\vecphy{p}\\ &=\vecphy{F}t \end{align*} $$$

Newton's third law of motion states that forces always occur in equal and opposite pairs.

A common way of stating Newton's third law of motion is that for every **action there is an equal and opposite reaction**. 'Action' and 'reaction' are paired, opposing forces (not a chain of events).

Newton's third law means that if an object A exerts a force on an object B, then object B exerts a force of equal magnitude and opposite direction on object A.

The gravitational force of the Earth pulls the moon. But the moon also pulls the Earth.

The forces of action and reaction ** act on different bodies ** and therefore do not automatically "cancel" each other out although they are of the same magnitude and opposite direction.

When a horse pulls a cart, the cart also exerts a force on the horse. To move forward, the horse must exert a forward force greater than the reverse force exerted on the horse by the cart.