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  • Revision:Kinematics Of A Particle

TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Kinematics of a Particle

These notes are based on the requirements of the M1 A Level mathematics module.

This is essentially the basic study of particle motion, and how certain quantities affect other quantities in a moving body. To be honest, the most important thing is the use of equations in these questions and knowing quickly which equation you should use.


SUVAT and statics

The most important word for this chapter is SUVAT, which stands for:

  • S (displacement),
  • U (initial velocity),
  • V (final velocity),
  • A (acceleration) and
  • T (time)

of a particle that is in motion.

Below is a list of the equations you MUST memorise, even if they are in the formula book, memorise them anyway, to ensure you can implement them quickly. Remember that time is everything in an exam.

\displaystyle v = u + at, which can be: \displaystyle a = \frac{v - u}{t}

\displaystyle v^2 = u^2 + 2as, which can also be: \displaystyle a = \frac{v^2 - u^2}{2s}

Remember that when you calculate the acceleration, it CAN be negative. This is called retardation, and is the slowing of an object due to resultant force opposing motion.

\displaystyle s = ut + \frac{1}{2}a(t^2)

Very useful for forming a quadratic in time and calculating using the formula. This equation can also be used quite effectively when calculating the time taken for something that is moving upwards (and decelerating) until it reaches zero velocity and then the time it takes to fall back to its original position. This is done by taking s as 0, since the particle has not effectively been displaced when it returns to its original position. Very useful.

When considering motion of an object in FREE-FALL this means that gravity is either accelerating or decelerating the object. To prevent confusion in kinematics, ALWAYS select whether the upward, or downward direction is positive. That way when using the gravitational acceleration constant you will not get peculiar answers. Even when dealing with objects on a surface, the use of direction to indicate positive or negative motion is helpful (for you and the examiner!) Always be sure to define which direction is positive.

Speed-time graphs

Two essential principles to be remembered.

The DISPLACEMENT is reflected by the TOTAL AREA UNDER THE GRAPH. So taking the area between two different times will give you the distance (or displacement) travelled in that time. The gradient of the graph, at any given point indicates the acceleration of the body. Much like other graphs the gradient can be taken as the change of y with a given change in x.

\displaystyle m = a = \frac{dx}{dy}.

When you have to calculate something, like the time where two different moving things will pass each other, you must deduce an equation for the displacement of each individual and their velocities (or accelerations). This equation must be in terms of T for each of the bodies and you should state that they must equal each other, since displacement will be equal when two people have moved the same distance. You then solve to find the value of T, which will be the point at which the two pass.

Sketching speed-time graphs

Simple I know, but always remember to have speed on your y axis and time as the x axis. Indicate constant acceleration by a straight, diagonal line going UP for ACCELERATION and DOWN for DECELERATION. Straight lines that are horizontal indicate constant velocity over a given time period.

Always ensure that the separate stages in motion are labelled on the axes. For example if there is constant acceleration for the first 5 seconds followed by 10 seconds of constant deceleration back to rest; then indicate this by having the points 5 and 15 marked on the x axis and draw a dotted line to show the change in motion upward from the point x = 5. Always remember to label axes, x and y, with important values indicated and units on the axes too! For example, you should note the maximum speed on the y axis and the end time on the x axis – this will help with calculations.


Originally written by RobbieC on TSR forums.

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