A.1 Kinematics

IB recommended length: 9 hours

Kinematics is the branch of science dedicated to describing the motion of objects without reference (yet) to the forces that cause the motion or the energy transfers involved in motion. In this section we will describe the motion of objects qualitatively (using words) and quantitatively (using numbers and equations). We will also make predictions about the future (and previous) position of an object can be predicted and how this motion analysis can be applied to solve real-world problems.

As this is the introductory topic we will also look at some of the basic technical and mathematical knowledge and skills required to study physics at this level.

Measuring position and time, t

In order to describe and analyse the motion of an object we need to be able to measure position, change in position and time. All measurements need units and everyone should agree on the definition of those units. In science we use an integrated system of measurements and units called SI (from the French Système international d'unités). For kinematics the base units we need to consider are the metre (m) and the second (s).

Sometimes the lengths we want to measure are much bigger or smaller than a metre, or times much longer or shorter than a second. There are three options in this case:

1. Use scientific notation (also known as standard form): For example, writing 5.2 × 10⁶ m, rather than 5 200 000 m.

2. Apply and use appropriate SI prefixes: For example, writing 5.2 megametres (Mm), rather than 5 200 000 m

3. Use a different, non-SI, unit: This is common for time (minutes, hours or days, etc.) but is occasionally used in distances (light-year, parsec)

In all cases we need to carefully consider the number of decimal places or significant figures we choose when writing numbers. In science we are always limited by the accuracy and precision of our measuring tools or how we use those tools. For example, a stopwatch can usually measure to the nearest hundredth of a second, but if we are using a stopwatch to time an event we are normally limited by human reaction times to ±0.2 s or ±0.3 s at best.

Distance and displacement, s

The separation of any two points in space is called the distance between them. Distance has the symbol d or the symbol x. Distance is measured in metres and is a scalar quantity. It has a size, called the magnitude and a unit, but nothing else. Displacement, on the other, is defined as a change in position. Displacement has a symbol s. The unit of displacement is the same as that of distance, the metre (m), but displacement is a vector quantity. 

A vector quantity means that as well as the magnitude of the measurement we also need to consider the direction, measured from the start of the displacement to the end. This can be described in a number of ways. In this section we will normally only worry about one-dimensional (1D) or two-dimensional (2D) movement.  In the case of 1D movement, movement along a line, we can use positive (+) and negative (−) signs to indicate direction. In the case of 2D movement we might define an angle from a known direction (e.g. 20º above the horizontal) or a cardinal direction (such as east or north). Understanding and working with vectors is an early mathematical requirement in the physics course.

Speed and velocity, v

The rate of change of the position of an object is its speed or velocity. Speed is the scalar form of the measurement, when we are not concerned about direction. To calculate the speed of an object we divide the distance (in metres) it travels, along the path it is travelling, by the time (in seconds) for which it is travelling:

speed = distance ÷ time      or     speed = distance / time

The SI derived unit of speed is metres per second (m/s). Alternative units, such as kilometres per hour (km / h) can be used, but SI units are generally preferred when possible. We can distinguish between average speed and instantaneous speed. The average speed is the found using the method above: total distance / total time taken. Instantaneous speed is the speed at a particular moment in time, the speed that would be measured by the speedometer in a car.

Apart from using a speedometer another way of finding average and/or instantaneous speed is to plot a distance-time graph, with time on the horizontal axis. On this graph speed is the gradient. The gradient of a line drawn from the origin to the final distance/time point will give the average speed. The gradient of the line at any point in time (maybe found using a tangent) gave the instantaneous speed.

Velocity is the vector equivalent of speed. If speed is velocity divided by time, then velocity is displacement divided by time.

velocity = displacement / time     or     v = s / ∆t

where displacement is in meters (m), time is in seconds (s) and thus velocity is in metres / second (m/s). This unit is the same as speed, the difference is that, as a vector, we need to consider a direction as well. As with displacement this could be done with a positive (+) or negative (−) sign in a one-dimensional situation, or using angles or other direction indications in two-dimensional situations.

Like speed, velocity also has average and instantaneous values. Since displacement is a vector quantity, a journey that ends in the place it begins has a total displacement of zero. Since velocity is displacement divided by time then the average velocity for this journey is zero, no matter how much time the journey takes.

In many questions the velocity at the start of the question may be different to the velocity at the end of the question. In these cases we use the symbol u to indicate initial velocity, v to indicate final velocity and ∆v to indicate the change in velocity, with ∆v = v - u.

Acceleration, a

Acceleration is the rate of change of speed or velocity (generally we will say velocity, but depending on circumstances we could be referring to speed).  Acceleration has the symbol, a, and has units of metres per second per second, or m/s/s, normally written as m/s².

a = ∆v / ∆t

where ∆v is the change in velocity (in m/s) and ∆t is the time taken (in seconds).

Acceleration is a vector quantity, direction should be indicated, and can have average and instantaneous values.

Equations of constant acceleration

In situations where the acceleration doesn't change during the situation / question we can use a set of equations connecting displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t). From the symbols involved these equations are also known as suvat equations.

• s = ((u+v)/2)t : displacement = average velocity × time

• v = u + at : final velocity = initial velocity + acceleration × time

• s = ut + 0.5 × at²

• v² = u² + 2as

Graphs of motion

We earlier described how a graph of distance or displacement against time (an s-t graph) can be used to find the average or instantaneous velocity by finding the gradient. Similar graphs can be used to analyse motion in the following ways:

• Displacement-time graphs: the gradient can be used to find average or instantaneous velocity (as described above)

• Velocity-time graphs: the gradient can be used to find average or instantaneous acceleration. The area under the graph (more correctly, the area between the line and the horizontal axis) can be used to find the displacement.

• Acceleration-time graphs: the area under the graph (more correctly, the area between the line and the horizontal axis) can be used to find the change in velocity

Projectiles

A projectile is any object moving through the air, moving only under the influence of gravity and air resistance. Air resistance we will ignore for the moment, and gravity we will deal with in more detail later. For now we should now that all objects moving through the air under the influence of gravity will experience a downwards acceleration, g of 9.81 m/s². Since the acceleration doesn't change we can use the suvat equations. 

One dimensional projectiles: the most trivial example of projectile motion is an object falling from a height. If the object is initially at rest then u = 0 m/s. If the projectile has an initial velocity up or down then the value of u will be non-zero and the sign will depend on the direction. An initial velocity down will have have the same sign as the acceleration due to gravity. An initial velocity  up will have a different sign to the acceleration due to gravity.

Two dimensional projectiles: To consider the vector motion of projectiles in two dimensions we need to separate the motion of the projectile into its horizontal and vertical components. If the initial velocity is known in terms of magnitude (u, in m/s) and angle θ with respect to the horizontal, then we can easily resolve into vertical ( uv = u sin θ ) and horizontal ( uh = u cos θ ) components. What unifies the two components is that the time taken to hit the ground, t in seconds, will the same for both. The vertical motion can be treated as the one-dimensional motion above, with acceleration towards the ground, g of 9.81 m/s². The horizontal motion is simpler. Since there is no horizontal acceleration the equations of constant motion can be used with a = 0 m/s². If needed to answer the question, the final, impact, velocity and angle of the projectile can be found by recombining the vectors using vector addition.

Fluid resistance: A fluid is any substance that can flow, i.e. liquids and gases. The affects of fluid resistance on projectile motion need only be considered qualitatively, instead of quantitatively. The fluid will slow the projectile down (negative acceleration) in the direction of motion, increasing the time of flight. The trajectory of the projectile will be changed, it won't go as high or as far, decreasing the range of the projectile. One consequence of introducing a fluid is that the acceleration due to gravity will be reduced as the projectile falls further and faster, until eventually (if the projectile falls far enough) it will be reduced to zero m/s² and the object will be falling at terminal velocity.

There is an excellent PhET simulation - including experimental error - on projectile motion.

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