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Uniform Rectilinear Motion (MRU)
Definition
Uniform Rectilinear Motion (MRU) is motion in a straight line at constant velocity. Characteristics:- Motion occurs in a straight line (rectilinear)
- Velocity is constant (does not change)
- Speed is constant (magnitude of velocity doesn't change)
- No acceleration (a = 0)
- No change in direction
Ejemplo: A car traveling on a straight highway at exactly 60 km/h (not accelerating or decelerating, and maintaining the same direction) is in uniform rectilinear motion.
Conditions for Uniform Rectilinear Motion
For motion to be uniform and rectilinear:
- Constant velocity: The velocity vector remains constant
- Zero acceleration: a = 0 m/s²
- Straight path: No change in direction
- Equal distances in equal times: The object covers the same distance in each time interval
Fundamental Equation of MRU
The fundamental equation relating position, velocity, and time is:
\[x = x_0 + vt\]
Where:
- x = final position (meters)
- x₀ = initial position (meters)
- v = constant velocity (m/s)
- t = time elapsed (seconds)
\[\Delta x = v \cdot \Delta t\]
Where Δx is displacement.
Deriving the Equation
Since velocity is constant:
\[v = \frac{\Delta x}{\Delta t} = \frac{x - x_0}{t - t_0}\]
If t₀ = 0 (starting from time zero):
\[v = \frac{x - x_0}{t}\]
Solving for x:
\[x = x_0 + vt\]
Solving MRU Problems
Example 1: Finding Position
Problem: A car starts at position x₀ = 10 m and travels at a constant velocity of v = 5 m/s. Where is the car after 8 seconds?\[x = x_0 + vt\]
\[x = 10 + (5)(8)\]
\[x = 10 + 40\]
\[x = 50 \text{ m}\]
The car is at position 50 m.
Example 2: Finding Velocity
Problem: An object moves from position 20 m to position 80 m in 10 seconds at constant velocity. What is the velocity?\[v = \frac{\Delta x}{\Delta t} = \frac{80 - 20}{10} = \frac{60}{10} = 6 \text{ m/s}\]
Example 3: Finding Time
Problem: A cyclist travels at 4 m/s. How long does it take to travel 200 m?\[\Delta x = v \cdot t\]
\[200 = 4 \cdot t\]
\[t = \frac{200}{4} = 50 \text{ seconds}\]
Example 4: Finding Displacement
Problem: A runner maintains a constant speed of 8 m/s for 25 seconds. How far does the runner travel?\[\Delta x = v \cdot t = 8 \times 25 = 200 \text{ m}\]
Graphical Representation of MRU
Position-Time Graph (x-t)
For uniform rectilinear motion, the position-time graph is a straight line.
Characteristics:- Linear relationship between position and time
- Slope = velocity (v = Δx/Δt)
- Steeper slope = faster motion
- Positive slope = motion in positive direction
- Negative slope = motion in negative direction
- Horizontal line = object at rest (v = 0)
Velocity-Time Graph (v-t)
For uniform rectilinear motion, the velocity-time graph is a horizontal line.
Characteristics:- Constant velocity (horizontal line)
- Area under curve = displacement
- No slope (acceleration = 0)
Comparing Different Motions
| Aspect | Object A | Object B | Object C |
|---|---|---|---|
| Velocity | 2 m/s | 5 m/s | 0 m/s |
| Position after 10 s (x₀ = 0) | 20 m | 50 m | 0 m |
| x-t graph | Gentle slope | Steep slope | Horizontal |
| v-t graph | Horizontal at 2 | Horizontal at 5 | Horizontal at 0 |
| Acceleration | 0 | 0 | 0 |
Real-World Applications
1. Transportation
- Cars on straight highways at constant speed
- Trains between stations
- Aircraft at cruising altitude
2. Astronomy
- Planets orbiting at relatively constant speeds
- Light traveling in vacuum at constant speed
3. Sports
- Athletes running at constant pace
- Swimmers maintaining constant speed
4. Engineering
- Conveyor belts moving at constant speed
- Elevators traveling between floors at constant velocity
Important Distinctions
Constant Speed vs Constant Velocity
Constant Speed: Only the magnitude remains the same- Object can change direction
- Speed is the scalar form
- Object travels in a straight line
- No acceleration
- Velocity is the vector form
Ejemplo: A car traveling around a circular track at 50 km/h has constant speed but NOT constant velocity (direction changes).
Uniform Motion vs Uniform Acceleration
Uniform Rectilinear Motion (a = 0):- Constant velocity
- Position changes linearly with time
- Horizontal v-t graph
- Velocity changes
- Position changes with time² (quadratic)
- Non-horizontal v-t graph
Key Takeaways
- MRU occurs in a straight line at constant velocity
- No acceleration in uniform motion (a = 0)
- Position equation: x = x₀ + vt
- Distance equals displacement in straight-line motion
- x-t graphs are straight lines (slope = velocity)
- v-t graphs are horizontal lines (zero acceleration)
- Area under v-t graph = displacement