Horizontal & Vertical Components (AQA A Level Maths: Mechanics)

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Dan

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Horizontal & Vertical Components

What is a projectile?

  • A projectile is a particle moving freely, under gravity, in a two-dimensional plane
  • Examples of projectile motion include
    • Sports such as basketball, javelin, archery, etc
    • Computer games such as Angry Birds

    2-6-1-horizontal-_-vertical-components-diagram-1

Modelling assumptions with projectiles?

  • The modelling assumptions with projectiles are
    • No air resistance, no horizontal forces
    • The projectile moves freely under gravity (no other vertical forces)
    • Ignore spin and the rotation of the projectile
    • Motion is symmetrical along the path of a parabola

What is the acceleration of a projectile?

  • As it is 2D there will be a horizontal component to acceleration (ax)and a vertical component (ay)
  • There will be no horizontal acceleration (ax = 0)
    • No forces are acting horizontally
    • Horizontal speed is constant

  • There will be constant vertical acceleration (due to gravity, ay = ± g)
  • ay may be positive or negative depending on which direction is taken as positive
  • The acceleration can be written as 2D vector
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What is the initial velocity of a projectile?

  • As it is 2D there will be a horizontal component to the initial velocity(ux) and a vertical component (uy)
  • A projectile is launched with initial speed U m s-1 at an angle θ to the horizontal
    • If it is projected below the horizontal then θ would be negative
  • Its initial velocity, u m s-1, is a vector with:
    • horizontal component, ux = U cosθ
    • vertical component, uy = U sinθ
  • The initial velocity can have a positive or negative vertical component as an object can be projected upwards or downwards

2-6-1-horizontal-_-vertical-components-diagram-2

Worked example

2.6.1_WE_Horizontal _ Vertical Components_1

A projectile is launched at an angle of 25° to the horizontal with speed 5 m s-1 .

Find the initial velocity, u m s-1 , of the projectile.

2-6-1-horizontal-_-vertical-components-example-solution

Examiner Tip

  • Sometimes they may give you the initial velocity as a vector already. To find the initial speed you would find the magnitude of the initial velocity and to find the angle of projection you could sketch a diagram and use trigonometry.
  • The horizontal speed is always constant so there will never be a point during the motion when the speed of the object is zero. Its minimum speed will be at its maximum height when its vertical velocity is instantaneously zero.

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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.