Log in
Mechanics & further mechanics (Topics 2 and 6)›Scalars, vectors and projectile motion (Topic 2B)

Scalars, vectors and projectile motion (Topic 2B)

Scalar and vector quantities, resolving and combining vectors, and independent projectile motion in Edexcel A-level Physics.
9 min

On this page

Scalar quantities have magnitude but no direction.

A table titled 'Scalar quantities' with two columns. The first column lists various scalar quantities: Distance, Speed, Mass, Time, Temperature, Energy, Power, Density, Work done, and Pressure. The second column provides descriptions for each quantity: Distance - How far an object travels (no direction), Speed - How fast an object moves, regardless of direction, Mass - Amount of matter in an object, Time - Duration of an event, Temperature - Measure of thermal energy or particle motion, Energy - Capacity to do work, Power - Rate of energy transfer, Density - Mass per unit volume, Work done - Energy transferred when a force moves an object, Pressure - Force per unit area (often treated as scalar at GCSE). The table is attributed to Medify.

An example of a scalar quantity is mass. The average mass of a human is which is just a number and has no direction.

Add to favourites

Vector quantities have both magnitude and direction.

Vector quantities and their descriptions: Displacement - The straight-line change in position, including both direction and magnitude. Velocity - Speed in a particular direction. Weight - Force due to gravity acting on a mass (acts towards the centre of Earth). Acceleration - Rate of change of velocity, including direction. Force - A push or pull acting in a specific direction. Momentum - Product of mass and velocity, direction same as velocity. Impulse - Change in momentum caused by a force acting over time. Drag / Frictional force - Resistive force acting opposite to motion. Lift - Upward force on an object in a fluid, opposite to weight. Magnetic field strength (field lines) - Has both magnitude and direction around magnets or currents.

An example of a vector quantity is force. A person of mass standing on the surface of Earth feels a force due to their weight with magnitude (where is the gravitational field strength) and direction pointing towards the Earth’s centre.

Add to favourites

Scalar quantities with the same units can be added or subtracted from each other.

For example, two rulers placed end to end have a total length:

Add to favourites

A vector can be represented visually by an arrow:

  • The length of the arrow is proportional to the vector magnitude.
  • The arrow points in the direction of the vector.
  • A vector is usually written with an arrow above the letter:
  • A vector can also be represented by an underlined letter:
A grid with horizontal and vertical lines. A green line extends horizontally from a green dot on the left to an arrow pointing right.

The diagram above shows a force vector drawn on paper with squares:

  • If the scale is then the force has a magnitude
  • The direction of the arrow indicates that the force is directed to the right.
Add to favourites

The diagram below shows the addition of two vectors and

  • Vectors can be added by placing the arrows end to end, as shown below.
  • The resultant vector can be found by drawing an arrow from the start of the first vector arrow to the end of the second arrow.
A graph with a grid background showing three vectors: a red vector labeled v→r, a green vector labeled v→1, and another green vector labeled v→2. The vectors originate from a green point at the bottom left corner of the graph.
Add to favourites

The diagram below shows the subtraction of two vectors from

  • To subtract one vector from another, treat it as adding the negative: . This means reversing the direction of to get , then adding it to end-to-end.
  • The resultant vector runs from the tail of the first vector to the tip of the reversed vector . It represents the difference between the two original vectors, both in magnitude and direction
A graph with a grid background showing two vectors. The first vector, represented by an arrow labeled →v1, is green and moves horizontally to the right. The second vector, represented by an arrow labeled →vr, is red and moves diagonally downward to the left. There is also an arrow labeled −→v2, which is green and moves vertically downward.
Add to favourites

The resultant of any two coplanar vectors can be determined by a scale drawing.

The addition of two coplanar displacement vectors is drawn in the example below:

  • For a scale, the magnitude of the resultant vector can be measured by a ruler as
  • The angle of the resultant vector to the horizontal is measured as
A graph showing vectors with labels: v1, v2, and vr. The angle θ is indicated at the origin where the vectors originate. The grid background is marked with horizontal and vertical lines.
Add to favourites

The magnitude of the resultant vector, can be found from Pythagoras’ theorem:

Where and are the magnitudes of the two perpendicular vectors.

The diagram below shows the vector addition of two perpendicular displacement vectors.

A graph showing vectors in a coordinate system. The red vector is labeled with v_r, and it points diagonally upwards. The green vector is labeled with v_1 and v_2, pointing horizontally and vertically, respectively. An angle θ is indicated at the origin.

In the diagram above, and so:

Add to favourites

Trigonometric relationships can be used to determine the direction of a resultant vector that is formed by two vectors and acting perpendicularly to one another.

For the diagram below, the angle, of the resultant vector to the horizontal can be found from:

A graph showing two vectors. The red vector labeled v_r points diagonally, with v_2 = 5 m at the top right and v_1 = 3 m at the bottom left. An angle θ is indicated at the origin.

In the diagram above is opposite the angle while is adjacent. Therefore, the angle can be calculated as follows:

It is useful to note that the magnitude and direction of any resultant vector can also be calculated for any coplanar vector using the cosine rule and the sine rule.

Add to favourites

A vector can be resolved into its perpendicular components.

A force acting in the plane may be resolved into its and components. For a force with magnitude pointing at an angle to the X axis:

  • The horizontal component magnitude is
  • The vertical component magnitude is
A graph showing a vector in a Cartesian coordinate system. The vector is represented by a red arrow labeled F with its magnitude |F| = F. The angle θ is indicated at the base of the vector. The horizontal component of the vector is labeled Fx = F cos θ, and the vertical component is labeled Fy = F sin θ. The y-axis is vertical and the x-axis is horizontal.

There are many contexts where resolving a vector into its perpendicular components is useful, often when an object is constrained to move in one direction or when only one direction of the vector is relevant:

  • An example of this is in foot races, the wind velocity component parallel to the track must be calculated to determine the headwind or tailwind during a race.
  • Another example of this is projectile motion, in which an object is acted on by gravity so that its horizontal velocity component remains the same (ignoring air resistance) but its vertical velocity component varies.
Add to favourites

A projectile is any object that is launched with an initial velocity and then moves solely under the influence of gravity in a uniform gravitational field. In the context of projectile motion, no other forces act on a projectile, such as thrust or air resistance.

A diagram illustrating projectile motion with a curved path. The horizontal axis is labeled 'Range'. An angle θ is shown at the starting point of the projectile, with an initial velocity 'u' indicated by an arrow. The vertical component of velocity is labeled 'Vy = 0 m s⁻¹'.

In projectile motion, horizontal and vertical velocity components can be considered independently:

  • The horizontal velocity of a projectile remains constant throughout its flight, as there is no horizontal acceleration (assuming air resistance is negligible).
  • The vertical velocity changes over time due to the acceleration caused by gravity.

The independent components combine to form a parabolic trajectory characteristic of projectile motion.

Add to favourites

For a projectile launched at an angle, it is useful to resolve its velocity into horizontal and vertical components using trigonometry.

A diagram showing a right triangle with the vertical axis labeled 'R sin θ' and the horizontal axis labeled 'R cos θ'. An angle θ is indicated at the bottom left corner, and a vector u is shown pointing diagonally upwards to the right.

For a projectile launched with an initial velocity at an angle to the horizontal:

  • the horizontal component of velocity is
  • the vertical component of velocity is .
Add to favourites

The initial vertical velocity of a projectile – equal to – determines the maximum height reached and the time of flight.

The initial horizontal velocity – equal to – affects the horizontal range (how far the projectile travels).

Two graphs showing the effect of changing vertical velocity and the effect of changing horizontal velocity. The top graph has a vertical axis labeled 'y' and a horizontal axis labeled 'x', with three curves in green, blue, and red representing different vertical velocities. The bottom graph also has a vertical axis labeled 'y' and a horizontal axis labeled 'x', with three curves in green, blue, and red representing different horizontal velocities.
Add to favourites

Question walkthrough

Resolving Velocity into Horizontal and Vertical Components

Resolve a projectile's initial velocity into horizontal and vertical components using trigonometry.

Key terms in projectile motion:

  • Time of flight : This is the total time a projectile remains in the air, from the moment it’s launched until it hits the ground. The time of flight is determined solely by the initial vertical velocity.
  • Maximum height : The maximum height is the highest point the projectile reaches, where its vertical velocity momentarily becomes zero before it starts descending. The maximum height is determined by the initial vertical velocity.
  • Range : The range is the horizontal distance the projectile travels from the launch point until it lands. The range depends on the initial horizontal velocity and the time of flight.
A diagram illustrating projectile motion with labeled components: 'u' representing initial velocity, 'usin θ' indicating the vertical component, 'ucos θ' indicating the horizontal component, 'max height, H' showing the peak height, and 'Range, R' denoting the horizontal distance traveled.
Add to favourites

Three common scenarios in projectile motion:

  • Vertical projection: The projectile is launched straight up, so there is no horizontal motion. Gravity directly opposes the vertical motion, slowing the projectile until it reaches its highest point, then pulling it back down.
  • Horizontal projection: The projectile is launched horizontally with no initial vertical velocity from an initial height above the ground. Its vertical motion begins only because gravity pulls it downward.
  • Projection at an angle: The most common scenario, where the initial velocity has both horizontal and vertical components. In this case, the velocity can be resolved into separate horizontal and vertical components, with gravity affecting only the vertical component.
Add to favourites

Useful equations for a projectile with an initial velocity , at an angle , include:

A table displaying three physics formulas: 'Time of flight' with the formula T = 2u sin(θ) / g, 'Maximum height' with the formula H = (u sin(θ))² / 2g, and 'Horizontal range' with the formula R = 2u² sin(2θ) / g. © Medify

These equations are valid only for a projectile that returns to its initial launch height upon landing.

Add to favourites

Question walkthrough

Time of Flight for Same-Height Projectile Launch

Calculate the total time of flight for a projectile launched and landing at the same height.

Question walkthrough

Maximum Height of a Projectile at an Angle

Calculate the maximum height reached by a projectile launched at an angle, using its vertical component of velocity.

Question walkthrough

Time of Flight from Projectile Range

Work backward from a projectile's range and launch angle to find its initial speed and total time of flight.

Learn how the vertical and horizontal components of velocity change (or don’t change) in projectile motion.

A graph showing a curved line with arrows indicating direction. Red arrows point upwards and downwards at various points along the curve, while blue arrows point horizontally to the left and right.
Do

Remember that the vertical velocity of an object decreases with time at a constant rate during projectile motion.

The vertical velocity decreases until it reaches zero at the highest point before reversing direction.

The vertical velocity then continues to decrease at a constant rate until it returns to the surface.

The horizontal component of velocity remains constant throughout the entire motion.

A graph of a curve with black points along it. There are red arrows pointing upwards and downwards, and blue arrows pointing left and right, indicating directions along the curve.
Don't

Assume the projectile maintains the same speed throughout its flight: gravity continuously affects vertical motion.

Assume that the horizontal component of velocity decreases over time during projectile motion.

Projectile motion always ignores any effects due to drag.

Add to favourites

Apply trigonometry correctly to resolve the horizontal and vertical components of an object’s trajectory in projectile motion.

SOH CAH TOA is a helpful mnemonic to remember the trigonometric ratios for right triangles:

  • ,
  • ,
  • .
A diagram showing a right triangle with a hypotenuse labeled 'u' in red. The vertical side is labeled 'usin θ' in blue, and the horizontal side is labeled 'ucos θ' in blue. There is a dashed line indicating the right angle.
Do

Use cos(θ) for the horizontal component and sin(θ) for the vertical component when resolving velocity, where θ is measured from the horizontal.

A diagram illustrating a right triangle with a hypotenuse labeled 'u' in red. The vertical side is labeled 'ucos θ' in blue, and the horizontal side is labeled 'usin θ' in blue. A dashed line outlines the triangle.
Don't

Mix up sine and cosine: this will lead to incorrect calculations for range, height, and time of flight.

Add to favourites

Question walkthrough

Time, Height and Range for an Elevated Launch

Calculate the time of flight, maximum height, and horizontal range for a projectile launched at an angle from an elevated platform.