Forces and Motion

Vectors

Vectors represent quantities with magnitude and direction. They can be added and subtracted using standard methods. Vectors can also be resolved into components:

Perpendicular components of a force act independently. This is the principle later used for SUVAT equations.

Displacement and velocity

Displacement is a vector quantity that represents an object’s distance and direction away from its starting point. The related scalar is distance, which represents the total length of the path travelled.

Velocity is a vector quantity that represents how fast and in what direction an object is moving. The related scalar is speed.

Displacement - time graphs The gradient of a displacement-time graph gives velocity.

Velocity-time graphs The gradient of a velocity-time graph gives acceleration. The area under a velocity-time graph gives displacement.

SUVAT

There are 5 SUVAT equations for motion with constant acceleration (the third is rarely ever used):

$$\begin{array}{rl}v& =u+at\\ s& =ut+\frac{1}{2}a{t}^2\\ s& =vt-\frac{1}{2}a{t}^2\\ s& =\frac{1}{2}(u+v)t\\ 2as& =v^2-u^2\end{array}$$

Where $u$ is initial velocity, $v$ is final velocity, $s$ is displacement, $t$ is time, and $a$ is acceleration.

Average velocity with constant acceleration is given by $\frac{v+u}{2}$.

Projectile motion

For projectile motion, the idea is to apply SUVAT separately for both the horizontal and vertical directions, and using the fact that the two times are equal; when an object reaches the ground vertically, it stops moving horizontally.

The time to maximum height (where the vertical velocity is zero) is exactly half the total time of flight. Alternatively, the parabolic trajectory is symmetrical about the midpoint at maximum height.

Iterative Models

Iterative models work by going step-by-step, simulating the behaviour of an object.

• An advantage is that iterative models can be used for when there are many variables e.g. air resistance increases as velocity increases.
• A disadvantage is that they do not accurately model reality, as they work in discrete steps, assuming that some variables do not change during each step. The accuracy of an iterative model can be increased by reducing the size of each time step.

Newton’s Laws of Motion

Newton’s laws of motion are:

• An object at rest will remain at rest, and an object in motion will remain in motion, unless acted on by an external force.
• $F=\frac{\Delta p}{\Delta t}=\frac{\Delta (mv)}{\Delta t}=m\frac{\Delta (v)}{\Delta t}=ma$ (force is also rate of change of momentum).
• For every action, there is an opposite and equal reaction. This is equivalent to the principle of conservation of momentum.

Momentum

Momentum is a vector quantity. It is the product of mass, $m$, and velocity, $v$.

$$p=mv$$

For any interaction in a closed system (elastic or inelastic), the principle of conservation of momentum holds: the total momentum before an interaction is equal to the total momentum after.

In elastic collisions, kinetic energy is conserved, whilst in inelastic collisions, kinetic energy is not conserved, and some kinetic energy is transferred to heat, sound, or deformation.

Impulse, $\Delta p$, is the change in momentum. It has units of $\text{Ns}$ (newton seconds), or ${\text{kgms}}^{-1}$ (kilogram metres per second). By $F=\frac{\Delta p}{\Delta t}$, $\Delta P=F\Delta t$. Impulse is also given by the area under a force-time graph.

Work

Work is the result of applying a force over a distance (in the line of action of the force). The work done is given by:

$$W=F\Delta s=Fs\cos \theta$$

Where $W$ is work done ($\text{J}$), $F$ is force ($\text{N}$), $\Delta s$ is distance in the line of action of the force ($\text{m}$), and $\theta$ is the angle between the line of action of the force and the displacement.

The $\cos \theta$ term corrects for distance not in the line of action of the force. This is an application of the dot product.

Energy

The principle of the conservation of energy states that energy cannot be created or destroyed, and can only be transferred from one store to another.

Energy is the capacity to do work. For motion, there are two energy equations:

$$E=mgh=\frac{1}{2}m{v}^2$$

Power is the rate at which work is done, or the rate at which energy is transferred:

$$P=\frac{\Delta E}{\Delta t}=\frac{Fs}{t}=Fv$$