<h3>The Main Idea</h3> <p><strong>Distance = Rate × Time</strong> is your universal tool for any motion problem.</p> <p>The key is identifying what you want, what you have, and using the formula forward or backward.</p> <h3>The 3-Step Approach</h3> <ol> <li><strong>What do you want?</strong> — Read the question sentence FIRST. Are you solving for distance, rate, or time?</li> <li><strong>What do you have?</strong> — Pull the given values from the problem. Note the units.</li> <li><strong>What's the connection?</strong> — Apply D = R × T, working forward or backward.</li> </ol> <h3>Working Forward</h3> <p>When you have <strong>Rate</strong> and <strong>Time</strong>, multiply to find Distance.</p> <p>D = R × T</p> <h3>Working Backward</h3> <p>When you have Distance and one other value, <strong>divide</strong> to find what's missing:</p> <ul> <li><strong>Find Rate:</strong> R = D ÷ T</li> <li><strong>Find Time:</strong> T = D ÷ R</li> </ul> <p><em>Remember: It's not about what the numbers are — it's about what they mean.</em></p> <h3>Units Check</h3> <p>Make sure your units align:</p> <ul> <li>If rate is in <strong>mph</strong>, time must be in <strong>hours</strong></li> <li>If rate is in <strong>feet per minute</strong>, time must be in <strong>minutes</strong></li> </ul>
Practice This Goal<h3>The Main Idea</h3> <p><strong>Break the trip into parts, then combine.</strong></p> <p>When an object travels at different speeds or in multiple legs, handle each part separately, then add to get totals.</p> <h3>Multi-Leg Trips</h3> <ol> <li><strong>Identify each leg</strong> — different speeds or different segments</li> <li><strong>Calculate D and T for each leg</strong> using D = R × T</li> <li><strong>Add distances</strong> to get total distance</li> <li><strong>Add times</strong> to get total time</li> </ol> <h3>Average Speed — The Big Trap</h3> <p><strong>Average speed ≠ average of the speeds!</strong></p> <p>You MUST use:</p> <p style="text-align: center; font-size: 1.1em;"><strong>Average Speed = Total Distance ÷ Total Time</strong></p> <p>This is the most common mistake. Always find total D and total T first.</p> <h3>Upstream and Downstream</h3> <p>When traveling with or against a current (water) or wind:</p> <ul> <li><strong>Downstream (with current):</strong> Effective Rate = Still Water Speed + Current</li> <li><strong>Upstream (against current):</strong> Effective Rate = Still Water Speed − Current</li> </ul> <p><em>The current helps you going with it, slows you going against it.</em></p> <h3>Breaks and Stops</h3> <p>If there's a break during the trip:</p> <ul> <li><strong>Distance during break = 0</strong> (you're not moving)</li> <li><strong>Time still counts</strong> if asking for "total time including break"</li> </ul>
Practice This Goal<h3>The Main Idea</h3> <p><strong>Two objects, same timeframe — combine their rates.</strong></p> <p>When two objects travel for the same amount of time, you can work with their combined or relative rate instead of calculating separately.</p> <h3>Opposite Directions = ADD</h3> <p>When two objects start together and move in <strong>opposite directions</strong>, they're moving apart.</p> <ul> <li>Combined Rate = Rate A + Rate B</li> <li>The distance between them grows at the combined rate</li> </ul> <p><em>Think: One goes left, one goes right — the gap opens faster.</em></p> <h3>Same Direction = SUBTRACT</h3> <p>When two objects travel in the <strong>same direction</strong>, the gap between them changes at the difference of their rates.</p> <ul> <li>Relative Rate = Faster Rate − Slower Rate</li> <li>The distance between them changes at this relative rate</li> </ul> <p><em>Think: Both going right, but one faster — the gap changes slowly.</em></p> <h3>The Shortcut</h3> <p>Instead of calculating each distance and then adding/subtracting:</p> <ol> <li>Combine the rates first (add or subtract based on direction)</li> <li>Multiply by the shared time</li> <li>Get the answer in one step</li> </ol> <h3>Facing Each Other = Opposite</h3> <p>Two objects moving <strong>toward each other</strong> is the same as opposite directions — they're closing the gap, so ADD their rates.</p>
Practice This Goal<h3>The Main Idea</h3> <p><strong>Different start times = different time frames. Calculate individually first.</strong></p> <p>When two objects don't start at the same time, you can't simply add or subtract rates — the times don't match. Find each distance separately, then combine.</p> <h3>Head Starts</h3> <p>When one object starts before the other:</p> <ol> <li><strong>Calculate the head start distance:</strong> Rate × Head Start Time</li> <li><strong>This creates the initial gap</strong> between the objects</li> <li>Then figure out what happens after both are moving</li> </ol> <h3>Different Times = Calculate Separately</h3> <p>You <strong>cannot</strong> combine rates when objects travel for different durations.</p> <ul> <li>Find Distance A using A's rate and A's time</li> <li>Find Distance B using B's rate and B's time</li> <li>Then add or subtract based on direction</li> </ul> <h3>Catch-Up Problems</h3> <p>When a faster object is chasing a slower one:</p> <ol> <li>Find the initial gap (head start distance)</li> <li>Find the relative rate (faster − slower) — this is how fast the gap closes</li> <li>Time to catch up = Gap ÷ Relative Rate</li> </ol> <p><strong>Key insight:</strong> If the chaser is slower than the leader, they will <strong>never</strong> catch up!</p> <h3>Watch the Reference Point</h3> <p>Read carefully: Is the question asking for time from when A started, or from when B started?</p> <p>If A has a head start and you find how long B takes to catch up, you may need to add A's head start time to get the total time from A's start.</p>
Practice This Goal