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Horizontal projection

Horizontal projection

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Further kinematics

Vector methods with projectiles

Variable acceleration in one dimension

Differentiating vectors

Integrating vectors

Application of forces

Static particles

Modelling with statics

Friction with static particles

Statics of rigid bodies

Dynamics and inclined planes

Connected particles

Projectiles

Horizontal projection

Projectiles: Horizontal and vertical components

Projection at any angle

Equation of the trajectory of a projectile

Forces and friction

Resolving forces

Inclined planes

The coefficient of friction

Moments

Moments

Resultant moments

Equilibrium

Centre of mass

Tilting

Laminas

Forces and motion

Force diagrams

Forces as vectors

Forces and acceleration

Motion in two dimensions

Connected particles

Pulleys

Variable acceleration

Functions of time: Displacement, velocity, acceleration

Using differentiation to find velocity and acceleration

Maxima and minima problems

Using integration to find velocity and displacement

Constant acceleration formulae

Constant acceleration

Displacement-time graphs

Velocity-time graphs

Constant acceleration: Deriving formulae

Constant acceleration formulae: SUVAT

Vertical motion under gravity

Modelling in mechanics

Modelling assumptions

Quantities and units in mechanics

Working with vectors

Normal distribution

The normal distribution

The normal distribution: Finding probabilities

The inverse normal distribution function

The standard normal distribution

Finding the mean and standard deviation

Normal approximation to the binomial distribution

Hypothesis testing with the normal distribution

Conditional probability

Set notation

Conditional probability

Conditional probability in Venn diagrams

Probability formulae

Conditional probability: Tree diagrams

Correlation and hypothesis testing

Exponential models

Measuring correlation

Hypothesis testing for zero correlation

Hypothesis testing I

Hypothesis testing

Hypothesis testing: Finding critical values

One-tailed tests

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Binomial distribution

Probability distributions

The binomial distribution

Cumulative probabilities

Probability

Calculating probabilities

Venn diagrams

Probability: Mutually exclusive and independent events

Probability: Tree diagrams

Representation of data

Outliers

Box plots

Cumulative frequency

Histograms

Comparing data sets

Correlation

Linear regression

Measures of location and spread

Measures of central tendency: Mean, median, mode

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Variance and standard deviation

Data collection

Population and samples

Sampling techniques

Types of data

Vectors II

3D coordinates

Vectors in 3D

Solving geometric problems

Applications to mechanics: 3D vectors

Numerical methods

Locating roots

Iteration

Staircase and cobweb diagrams

Newton-Raphson method

Numerical methods: Applications to modelling

Integration II

Integrating standard functions

Integrating f(ax + b)

Integration with trigonometric identities

Reverse chain rule

Integration by substitution

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Finding areas using integration

The trapezium rule

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Differentiation II

Differentiating sin x and cos x

Differentiating exponentials and logarithmic functions

The chain rule

The product rule

The quotient rule

Differentiating inverse functions

Differentiating trigonometric functions

Parametric differentiation

Implicit differentiation

Second derivatives: Concave and convex functions

Connected rates of change

Trigonometry and modelling

Addition formulae

Using the angle addition formulae

Double angle formulae

Solving trigonometric equations: Double angle formula

Simplifying a cos x + b sin x

Proving trigonometric identities

Modelling with trigonometric functions

Trigonometric functions

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Graphs of sec x, cosec x and cot x

Solving equations with sec x, cosec x and cot x

Further trigonometric identities

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Radians

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Areas of sectors and segments

Solving trigonometric equations

Small angle approximations

Sequences and series

Sequences revision

Arithmetic sequences

Arithmetic series

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Geometric series

Sum to infinity

Sigma notation

Modelling with sequences and series

Binomial expansion II

Binomial expansion: (1+x)^n

Binomial expansion: (a+bx)^n

Binomial expansion with partial fractions

Binomial expansion of compound expressions

Parametric equations

Parametric equations

Using trigonometric identities

Sketching parametric curves

Points of intersection of parametric curves

Modelling with parametric equations

Functions

Functions and mappings

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The modulus function

y = |f(x)| and y = f(|x|)

Combining transformations

Solving modulus equations

Modulus inequalities

Algebraic fractions

Operations with algebraic fractions

Partial fractions

Repeated factors

Algebraic division

Proof II

Proof by contradiction

Criticising proofs

Vectors I

Vectors

Representing vectors

Magnitude and direction of a vector

Position vectors and displacement vectors

Solving geometric problems

Modelling with vectors

Integration I

Integrating x^n

Indefinite integrals

Finding functions by integrating

Definite integrals

Area under a curve

Area under the x-axis

Area between a curve and a line

Differentiation I

Finding the gradient of a curve

Differentiation from first principles

Differentiating x^n

Differentiating functions with multiple terms

Gradients, tangents and normals

Increasing and decreasing functions

Second order derivatives

Stationary points

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Modelling with differentiation

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Exponential functions

y = e^x

Exponential modelling

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Laws of logarithms

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Natural logarithms

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Trigonometric equations

Angles in all four quadrants

Exact trigonometric values

Trigonometric identities

Simple trigonometric equations

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Equations and identities

Trigonometry

The cosine rule

The sine rule

Area of a triangle

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Trig graphs: sin x, cos x, tan x

Transforming trigonometric graphs

Binomial expansion I

Pascal's triangle

Factorial notation

Binomial expansion

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Binomial estimation

Circles

Midpoints and perpendicular bisectors

Equation of a circle

Intersection of straight lines and circles

Using tangent and chord properties

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Intersection of two circles

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y = mx + c

Equations of straight lines

Parallel and perpendicular lines

Length and area

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Transformations

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Points of intersection

Using the discriminant

Direct and inverse proportion

Polynomials

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Expanding brackets

Factorising

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The factor theorem

Equations and inequalities

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Set notation and interval notation

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Solving disguised quadratics

Indices and surds

Laws of indices

Negative and fractional indices

Surds

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Proof I

Mathematical structure and arguments

Disproof by counterexample

Proof by deduction

Proof by exhaustion

Explainer Video

Tutor: Meera

Summary

Horizontal projection

In a nutshell

Projectile motion involves the study of how a particle moves under the influence of gravity. When considering the motion of a particle in 2D, the vertical and horizontal components are independent of each other, and therefore the calculations for vertical and horizontal movement can be considered separately. 


​

Horizontal projection

When considering the movement of a particle projected horizontally, consider the horizontal and vertical motion separately. Assume air resistance is zero and ignore any rotational effects of the particle.


Horizontal motion

There is no acceleration in the horizontal direction as the only acceleration is vertical, (acceleration due to gravity). Therefore, the horizontal component of a projectile is considered to have constant velocity. The equation s=vt\bold s= \bold v ts=vt is used.


Vertical motion

When projected horizontally, the initial vertical component of the velocity is 0 m.s−10 \ m.s^{-1}0 m.s−1. Acceleration due to gravity is 9.8 m.s−29.8 \ m.s^{-2}9.8 m.s−2, and so the formulae for constant acceleration can be used in the vertical direction.​


Equations

description

equation

Horizontal motion is modelled as having 
constant velocity (a=0)(a=0)(a=0).​
​s=vts=vts=vt​​

​Vertical motion is modelled as having 

constant acceleration due to gravity, 

so the constant acceleration formulae can be used.
​v=u+ats=(u+v)2ts=ut+12at2s=vt−12at2v2=u2+2as\begin{aligned}v&=u+at \\ \\s&=\dfrac{(u+v)}{2}t \\\\s&=ut+\dfrac{1}{2}at^2 \\\\s&=vt-\dfrac{1}{2}at^2 \\\\v^2&=u^2+2as\end{aligned}vsssv2​=u+at=2(u+v)​t=ut+21​at2=vt−21​at2=u2+2as​​​
​


Variable definitions

Quantity name

symbol

unit name

unit

​displacementdisplacementdisplacement​​
​sss​​
​metremetremetre​​
​mmm​​
​initial velocityinitial \ velocityinitial velocity​​
​uuu​​
​metres per secondmetres \ per \ secondmetres per second​​
​m.s−1m.s^{-1}m.s−1​​
​final velocityfinal \ velocityfinal velocity​​
​vvv​​
​metres per secondmetres \ per \ secondmetres per second​​
​m.s−1m.s^{-1}m.s−1​​
​accelerationaccelerationacceleration​​
​aaa​​
​metres per second squaredmetres \ per \ second \ squaredmetres per second squared​​
​m.s−2m.s^{-2}m.s−2​​
​timetimetime​​
​ttt​​
​secondssecondsseconds​​
​sss​​
​acceleration due to gravityacceleration \ due \ to \ gravityacceleration due to gravity​​
​ggg=9.8=9.8=9.8​​​
​metres per second squaredmetres \ per \ second \ squaredmetres per second squared​​
​m.s−2m.s^{-2}m.s−2​​



Example

A particle is projected horizontally at 20 m.s−120\ m.s^{-1}20 m.s−1​ from 80 m80\ m80 m above the ground. Find the time taken to reach the ground and the horizontal range.


First draw a sketch to show the motion of the particle:


For this question, take vertically up to be positive, and horizontally right to be positive.

To calculate the time taken, use the vertical motion. First state all the known vertical values:

​s=−80u=0v=?a=−9.8t=?s=-80 \qquad u = 0 \qquad v= ? \qquad a=-9.8 \qquad t=?s=−80u=0v=?a=−9.8t=?​

As vvv is unknown, use s=ut+12at2s=ut+\dfrac 1 2 at^2s=ut+21​at2 to calculate ttt​:

−80=0+12×−9.8×t2−80=−4.9t2t2=16.33t=4.04 s (3 s.f.)\begin{aligned}-80&=0+\cfrac{1}{2} \times -9.8 \times t^2\\-80&=-4.9t^2\\ t^2&=16.33\\t&=4.04 \ s\thinspace (3\thinspace s.f.)\end{aligned}−80−80t2t​=0+21​×−9.8×t2=−4.9t2=16.33=4.04 s(3s.f.)​​​

The time taken to reach the ground is 4.04 s‾\underline{4.04 \ s}4.04 s​.


To calculate the horizontal range, use s=vts=vts=vt:

s=20×4.04s=80.8 (3 s.f.)s=20\times 4.04\\ s=80.8 \ (3\thinspace s.f.)s=20×4.04s=80.8 (3s.f.)​​

The particle has a horizontal range of 80.8 m‾\underline{80.8\ m}80.8 m​.


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Equations of motion

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Equations of motion

Projectile motion

Unit 2

Projectile motion

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Horizontal projection

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Horizontal projection

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FAQs - Frequently Asked Questions

How do you work with projectile motion in 2D?

When considering the motion of a particle in 2D, the vertical and horizontal components are independent of each other, and therefore the calculations for vertical and horizontal movement can be considered separately.

Does the horizontal speed change during projectile motion?

There is no acceleration in the horizontal direction as the only acceleration is vertical, (acceleration due to gravity). Therefore, the horizontal component of a projectile is considered to have constant velocity.

What is projectile motion?

Projectile motion involves the study of how a particle moves under the influence of gravity.

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