3.5 Solve Uniform Motion Applications

Uniform motion applications help us understand how objects move at constant speeds. We use the distance, rate, and time formula to calculate travel times, speeds, and distances in real-world situations like road trips and races.

Diagrams and tables make it easier to visualize and organize motion data. We can compare speeds, analyze different scenarios, and solve problems involving movement. These skills are useful for planning trips, estimating arrival times, and understanding basic physics concepts.

Solving Uniform Motion Applications

Distance, rate, and time formula

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• Formula [d = rt](https://www.fiveableKeyTerm:d_=_rt) calculates distance traveled ($d$) by multiplying rate or speed ($r$) and time ($t$)
• Rearrange formula to solve for rate $r = \frac{d}{t}$ (distance divided by time) or time $t = \frac{d}{r}$ (distance divided by rate)
• Ensure consistent units (convert hours to minutes, feet to miles) before calculating
• Apply formula to real-world scenarios (road trips, flights, running races)
• Example: Car drives 180 miles in 3 hours, so rate is $r = \frac{180}{3} = 60$ mph (velocity)

Diagrams and tables for uniform motion

• Visually represent motion using diagrams with arrows for direction, labels for start/end points, and annotations for distance, rate, time
• Organize data in tables with columns for distance, rate, time
  1. List known values
  2. Calculate missing values using distance, rate, time formula
  3. Fill in table completely
• Example diagram: Airplane flies 2,400 miles from New York to Los Angeles in 6 hours
• Example table: | Scenario | Distance (mi) | Rate (mph) | Time (hr) | |----------|---------------|------------|-----------| | Road trip| 240 | 60 | 4 | | Marathon | 26.2 | ? | 4.5 | | Flight | ? | 500 | 3.5 |

Speed comparisons in uniform motion

• Compare speeds of objects traveling same distance in different times
  1. Calculate each object's rate using $r = \frac{d}{t}$
  2. Object with higher rate is faster
• Example: Cyclist A rides 30 miles in 2 hours ($r_A = 15$ mph), Cyclist B rides 30 miles in 1.5 hours ($r_B = 20$ mph), so Cyclist B is faster
• For fixed distance, speed inversely proportional to time (shorter time means higher speed)
• Practical applications: races (running, swimming), comparing transportation methods (driving, flying), optimizing travel plans

Motion and Kinematics

• Kinematics: branch of physics dealing with motion of objects without considering forces
• Displacement: change in position of an object (vector quantity)
• Acceleration: rate of change of velocity over time
• Scalar quantities: have magnitude only (e.g., speed, distance)
• Vector quantities: have both magnitude and direction (e.g., velocity, displacement)