Distance-Time Graphs and Speed Calculation

Distance-Time Graphs

• Distance-time graphs display the relationship between distance traveled and the time taken.
• The horizontal axis represents time, and the vertical axis represents distance.
• A straight line on a distance-time graph indicates constant speed.
• The gradient of the line on a distance-time graph corresponds to the speed.
• A steeper line indicates a higher speed.
• A horizontal line indicates zero speed (stationary).
• The area under a speed-time graph represents the distance traveled.
• The distance covered can be calculated by measuring the vertical distance on the graph.
• Time is calculated by measuring the horizontal distance on the graph.
• Curves on distance-time graphs indicate changes in speed.

Calculating Speed from a Distance-Time Graph

• Speed is calculated by finding the gradient of the line on a distance-time graph.
• Gradient is calculated by finding the vertical rise over the horizontal run.
• The formula for speed is: speed = distance/time
• The units for speed derived are usually meters per second (m/s), kilometers per hour (km/h), or miles per hour (mph).
• The speed can be calculated over different time intervals.

Interpreting Distance-Time Graphs

• An upward sloping line represents movement away from the starting point.
• A downward sloping line represents movement towards the starting point.
• Stationary periods are shown by a horizontal line.
• Changes in speed are shown by different gradients on the graph.
• The starting point is when time = 0 on the graph.

Simultaneous Equations

• Simultaneous equations are a set of two or more equations with the same variables that need to be solved together.
• The goal is to find the values of the variables that satisfy both equations.
• Several methods are available to solve simultaneous equations.
• Two common methods are: substitution and elimination.

Solving Simultaneous Equations Using Substitution

• In the substitution method, one equation is rearranged to express one variable in terms of the other.
• Then, this expression is substituted into the second equation.
• This simplifies the second equation to a single variable equation, which can be easily solved.
• Once the value of one variable is found, it's substituted back into either of the original equations to calculate the value of the other variable.

Solving Simultaneous Equations using Elimination

• The elimination method aims to cancel out one of the variables by either adding or subtracting the equations.
• If the coefficients of one variable are opposites (e.g., +3x and -3x), adding the equations will eliminate that variable.
• If the coefficients of a variable are the same (e.g., 2x and 2x), subtracting the equations will eliminate the variable.
• After eliminating one variable, we end up with a single-variable equation, which can be solved for the value of the remaining variable.
• The value of the solved variable is substituted back into either the original equation to find the other variable.

Real-world Applications of Simultaneous Equations

• Simultaneous equations are used to solve various real-world problems.
• They are applied in various disciplines including economics, engineering, physics, and more.
• For instance, they can help determine the prices of two products based on given conditions.
• They are instrumental in modelling and analyzing systems involving multiple variables and constraints.
• They are fundamental in many mathematical and scientific formulations.

Relationships between Distance-Time Graphs and Simultaneous Equations

• Distance-time graphs illustrate situations that can be described using simultaneous equations.
• Imagine two objects moving at different speeds.
• Their positions can be determined using distance-time graphs.
• Determining the intersection points (where objects meet) involves a system of simultaneous equations relating distance, speed, and time.