Relative Motion Problems

• Relative motion problems involve understanding how velocities add or subtract depending on the direction of movement.

Basic Upstream and Downstream Motion

• When moving downstream, a boat's speed relative to the shore is the sum of its speed in still water and the river current's speed.
• When moving upstream, a boat's speed relative to the shore is the difference between its speed in still water and the river current's speed.
• To determine the time it takes to travel a certain distance upstream, divide the distance by the boat's upstream speed (speed in still water minus the current).
• Distance traveled upstream is calculated by multiplying the upstream speed (speed in still water minus the current) by the time traveled.

Crossing the River Perpendicularly

• When crossing a river, the downstream drift is the product of the river current's speed and the time it takes to cross the river.
• The time it takes to cross a river is determined by dividing the width of the river by the swimmer's speed in still water.
• Calculate a ferry's speed in still water using the Pythagorean theorem if it moves directly across a river with a current.
• A swimmer's speed can be determined by using the river width, downstream distance and current.

Minimum Time to Cross

• To minimize crossing time, a boat should travel in a direction perpendicular to the river's flow, regardless of the current.
• The minimum crossing time is calculated by dividing the width of the river by the boat's speed in still water.

Minimum Distance to Cross

• To reach a point directly opposite the starting position, a boat must steer at an angle upstream to counteract the river's current.
• The boat's velocity relative to the riverbank can be found using trigonometric functions, considering both the boat's speed and the river's current.
• The time it will take to cross can be found using the effective velocity.
• Determine the angle using inverse sine

Moving Observers on a Boat

• A person's velocity relative to the riverbank is the sum of their walking speed, the boat's speed, and the river current's speed, assuming they are moving in the same direction.
• Time to walk the boat can be found by dividing the length of the boat by the persons walking speed.

Chasing and Catching Up

• When a boat is chasing a log, the relative speed is the boat's upstream speed minus the river current's speed, which equals the boat's speed relative to the log.
• Calculate the time it takes to catch up by dividing the initial distance by the relative speed.
• For jet skis, the time until the first one catches up is found by dividing the starting distance by the difference in their speeds relative to the shore.

Advanced Relative Motion

• Using the width of the river and the distance downstream, calculate the time to cross.
• Divide the width of the river by the time to determine the boats speed.
• Add the river flow speed to the target speed to find overall speed.
• Divide the distance by the speed to determine the time.

River Crossings with Angles

• You can find the angle if you treat the values as a triangle.
• The adjacent is the kayakers speed.
• The opposite is the speed of the river.
• Use inverse sine to determine the angle.
• Multiply the river flow rate by the time to find the downstream distance.
• Divide the width by the kayaking speed to determine time.

Combination Problems

• Needs more information to calculate.
• Use vector addition.
• The swimmers will meet based upon a ratio.
• Determine separate speeds, add together, and then divide the distance by that value.