Calculating Speed

• The mathematical relationship for speed is derived from the ratio of distance to time. If an object covers more distance in the same amount of time, its speed is higher.

• For any object moving at a constant rate, the relationship is expressed by the formula:

$v = \frac{s}{t}$

• In this formula, $v$ represents speed ($m/s$), $s$ represents distance traveled ($m$), and $t$ represents the time taken ($s$).

• This principle assumes that the rate of motion is steady throughout the interval being measured.

Calculating Average Speed

• When speed is not constant, we calculate the average speed by dividing the total distance by the total time taken for the entire trip.

• Step 1: Identify the total distance traveled ($s_{total}$) and ensure it is in meters.

• Step 2: Identify the total time taken ($t_{total}$) and ensure it is in seconds.

• Step 3: Apply the formula: $v_{avg} = \frac{s_{total}}{t_{total}}$.

Using Light Gates

• In laboratory settings, light gates provide higher precision than stopwatches by removing human reaction time.

• A 'flag' of known length ($d$) attached to an object breaks a light beam. The timer measures the duration ($t$) the beam is blocked.

• The speed is calculated as $v = \frac{d}{t}$, where $d$ is the length of the flag.

• It is vital to distinguish between the speed at a specific moment and the speed over a duration.

• Scalar vs. Vector: Speed only accounts for how fast an object moves, whereas velocity accounts for both speed and the direction of travel.

• Unit Consistency: Always check the units provided. If distance is in kilometers and time is in hours, you must convert them to meters and seconds if the answer requires $m/s$.

• The Formula Triangle: Use a formula triangle with $s$ at the top and $v, t$ at the bottom to easily rearrange for distance ($s = v \times t$) or time ($t = s / v$).

• Sanity Checks: Evaluate if your calculated speed is realistic. For example, a person walking should be around $1.5\ m/s$; if your calculation results in $150\ m/s$, you likely missed a unit conversion.

• Graph Gradients: On a distance-time graph, a steeper line always indicates a higher speed. A horizontal line indicates the object is stationary (speed = $0$).

• Confusing Distance and Displacement: Speed is based on the total path length (distance), not the straight-line gap between start and end points (displacement).

• Incorrect Time Conversions: A common error is assuming $1.5$ hours is $150$ minutes. Always multiply hours by $3600$ to get seconds ($60 \text{ mins} \times 60 \text{ secs}$).

• Non-Zero Starts: When calculating gradients from graphs, always use the change in values ($\Delta y / \Delta x$) rather than just reading a single point's coordinates, especially if the line doesn't pass through the origin.