Motion Problems

Motion problems involve objects moving at constant speed and use relationships between distance, rate (speed), and time.

Key Formula:

$$ \text{Distance} = \text{Rate} \times \text{Time} $$

Variable Notation:

$d$ – distance traveled (in km, m, etc.)
$r$ – rate or speed (in km/hr, m/s, etc.)
$t$ – time taken (in hours, minutes, etc.)

Common Motion Scenarios:

Objects moving toward or away from each other
Round trips or return journeys
Chase problems (one faster object catching up)

Use the distance formula to write equations for each object or direction, then solve based on the relationship given in the problem.

Variable Speed

In problems involving rivers or flowing water, the motion of a boat or swimmer is affected by the speed of the current. The total effective speed depends on whether the motion is going with or against the current.

Key Relationships:

With the current: The object moves faster.
Against the current: The object moves slower.

Let the variables be:

$r$ = rate of the boat or swimmer in still water
$c$ = rate of the current

Effective Speeds:

With the current:

$$ \text{Effective Rate} = r + c $$

Against the current:

$$ \text{Effective Rate} = r - c $$

Once you know the effective rate, use the distance formula:

$$ \text{Distance} = \text{Rate} \times \text{Time} $$

This type of problem often involves comparing times or distances for round trips or upstream/downstream travel.

Problem 1:

Two pals A and B run at constant speeds along a circular track 1350m in circumference. Running in opposite directions they meet every 3 minutes, while running in the same direction they are together every 27 minutes. Determine their speeds in km/hr. Ans. A=15kph B=12kph

See images:

Problem 2:

An airplane travels 500km against the wind in one hour and 45 minutes. Travelling the same distance with the wind, the airplane consumed one hour and 15 minutes. Find the velocity of the wind in kph. Ans. 57.14kph

See images: