Basic kinematics

Understanding Motion

Motion is the change in position of an object over time. When an object moves, it changes its location relative to a reference point or coordinate system. Motion can be described in terms of how far something moves, how fast it moves, and whether its speed is changing.

Distance is the total length of the path traveled by an object, regardless of direction. It is always a positive value and is measured in units such as meters (m), kilometers (km), or miles. Distance tells us "how much ground was covered" during the motion.

Speed is the rate at which distance is covered over time. It tells us how fast an object is moving and is calculated as distance divided by time. Speed is measured in units such as meters per second (m/s) or kilometers per hour (km/h). Like distance, speed is always positive or zero.

Acceleration is the rate at which speed changes over time. When an object speeds up, slows down, or changes direction, it is accelerating. Acceleration is measured in units such as meters per second squared (m/s²). Positive acceleration means the object is speeding up, while negative acceleration (sometimes called deceleration) means it is slowing down.

Directed motion

In mechanics or kinematic questions most quantities are 'directed' - or have a direction. When you move a certain distance, you also move in a direction: up, down, left, right, North, South etc. The effect of your movement results from both how far you move from your start point and in which direction you moved. In order to determine your direction of movement you need some axes - more on that later.

Movement expressed as distance plus direction is called displacement. Speed plus direction is called velocity. Acceleration plus direction is, well, acceleration. Time does not have a direction in the sense of up, down, left, right but we can think of a positive time as being something happening in the future and a negative time as being something happening in the past.

Constant speed

When an object moves with constant speed, it covers equal distances in equal time intervals. The animation below demonstrates this concept - the circle travels at a steady rate from left to right, taking exactly three seconds to complete its journey.

Adding numbers to constant speed animation

This animation shows the same constant speed motion, but now with position (s) and time (t) labels appearing at key points. Notice how both position and time increase proportionally - at each second, the position increases by 1 meter.

Accelerating motion - falling particle

Unlike constant speed motion, when an object accelerates, it covers increasing distances in equal time intervals. This animation shows a particle falling under gravity, where the distance fallen increases with the square of time (s = ½gt²). Notice how the particle falls progressively farther during each second.

Accelerating motion - horizontal

The same acceleration pattern can occur in any direction, not just vertically. This animation shows the same accelerating motion as the falling particle, but now moving horizontally from left to right. The particle still covers increasing distances in equal time intervals (1, 4, 9 units at t=1, 2, 3).

Indicators of a constant acceleration or suvat question

Recognizing when a problem requires constant acceleration, or suvat, methods is the crucial first step in solving it. Here are the key indicators to look for:

Keywords: Look for words like "constant acceleration", "starts from rest", "the vehicle passes the start with velocity u ms⁻¹", etc.

Direction of motion: In simple (1D) problems the object such as a car or a ball is travelling in a straight line such as falling vertically or travelling along a horizontal surface. A particle travelling along a flat inclined plane is travelling in 'one' direction parallel to the plane but this requires resolving velocities (examples later).

Why diagrams are essential

Drawing a clear diagram is one of the most important steps in solving projectile motion problems. A good diagram helps you visualize the problem, identify what you know, and determine what you need to find.

Coordinate system: Always establish a clear coordinate system. Typically, the x-axis is horizontal and the y-axis is vertical, with the origin at a convenient location (often the launch point).

Initial position: Mark the starting position of the projectile clearly on your diagram.

Velocity components: Draw and label the initial velocity vector and its horizontal and vertical components \(v_{0x}\) and \(v_{0y}\).

Trajectory path: Sketch the parabolic path the projectile will follow.

Key points: Mark important points such as the maximum height, landing position, or any target locations mentioned in the problem.

Known and unknown quantities: Label all known values and indicate what you need to find.

Reference axes

Axes must have a start point, or origin. One axis must point in a straight line in one direction, the second axis must point in one of the directions perpendicular (at right angles) to the first axis. All directed quantities must have their direction expressed in relation to the axes.

Example:

Specific values

• The point from where we are measuring displacement and start time, the values \(s=0\) and \(t=0\) are assigned (1D case)
• Even if the particle starts moving from a position above the ground, say one metre, then \(s=0\) is usually measured from where object starts moving, not from the ground
• 'Starts from rest' means \(u=0\) or \(v_0 = 0\)

Example: a particle at rest is dropped from the top of a 5m tall tower to the ground below.

Notation

If a problem is about (uniform acceleration) motion in a straight line (it is one-dimensional), then we can use the following variables to express and solve our problem:

• \(s\) is displacement from origin
• \(u\) is initial velocity
• \(v\) is final velocity
• \(a\) is (constant) acceleration
• \(t\) is time (a non directed quantity)

If a problem is about (uniform acceleration) motion in a plane (it is two-dimensional), then we can use the following variables to express and solve our problem:

• \(s_x\) and \(s_y\) are displacement in the x and y direction
• \(u_x\) and \(u_y\) are initial velocity in the x and y direction
• \(v_x\) and \(v_y\) are final velocity in the x and y direction
• \(a_x\) and \(a_y\) are acceleration in the x and y direction

Sometimes we use the following notation:

• \(v_0\) is the initial speed when the particle is initially travelling at a non-zero angle relative to one of the reference axes
• \(v_{0x}\) and \(v_{0y}\) are initial velocity in the x and y direction

Constant acceleration equations

The five standard constant acceleration equations (also known as suvat equations) relate displacement, initial velocity, final velocity, acceleration, and time:

Where: s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time

In a kinematics question you will be given 3 variable values, often expressed in words like 'initially at rest' and be asked to determine the value of one other variable. So the question will be about 4 variables, three known values, one to be determined. Find the equation that relates these four variables and apply it to the values.

Problem-solving approach

Apply the method described on the Solution Method page.