Motion in a Straight Line
Motion is everywhere—a falling apple, a speeding car, a spinning electron. Yet describing motion precisely is surprisingly subtle.
Start with the simplest version: this lesson is about Motion in a Straight Line. If you can explain the core idea to a friend using everyday language, examples, and one clear reason why it matters, you have moved from memorising to understanding.
Motion is everywhere—a falling apple, a speeding car, a spinning electron. Yet describing motion precisely is surprisingly subtle. This chapter focuses on motion along a single line: straight-line motion. We'll learn to distinguish between speed and velocity, understand how acceleration changes motion, and derive the kinematic equations that predict where and when moving objects will be. These concepts form the foundation for understanding all motion in the universe.
Speed vs. Velocity: A Critical Difference
Your car's speedometer shows 60 km/h. That's your speed—how fast you're going, without regard to direction. But if you're driving north at 60 km/h, that's your velocity—speed with direction attached.
Speed is the total distance traveled divided by the time taken. It's always positive. Walk 5 meters forward and 5 meters backward, and your average speed is 10 m / time, even though you ended where you started.
Velocity is displacement (change in position) divided by time. It can be positive or negative, depending on your direction. Walk 5 meters north then 5 meters south, and your average velocity is zero—you're back where you started.
For motion in one dimension, we use a sign convention: positive direction (usually right or forward) and negative direction (left or backward). A velocity of -10 m/s means moving backward at 10 m/s.
Instantaneous vs. Average
When you glance at your speedometer, you see instantaneous speed—how fast you're going right now, at this instant. Over a long trip, your average speed is total distance divided by total time.
Mathematically, instantaneous velocity is the limit of velocity over infinitesimally small time intervals. It's the slope of the position-time graph at a single point. Average velocity is the slope of the secant line connecting two points on the position-time graph.
Acceleration: The Rate of Change
Acceleration measures how quickly velocity is changing. If your velocity changes from 0 to 10 m/s in 2 seconds, your acceleration is 5 m/s² (or 5 m/s per second). This means every second, your velocity increases by 5 m/s.
Like velocity, acceleration is a vector with direction. Acceleration doesn't have to mean speeding up—it means changing velocity. Slowing down is also acceleration (sometimes called deceleration or negative acceleration). Even if you're moving at constant speed along a curve, you're accelerating because your direction is changing.
The most important insight: acceleration is not velocity. An object can have zero velocity but non-zero acceleration (like a ball at the peak of its throw, momentarily at rest but still accelerating downward). An object can have constant velocity and zero acceleration, as a car on a straight, flat highway at constant speed.
Kinematic Equations: Predicting Motion
For uniformly accelerated motion (constant acceleration), we derive four kinematic equations that relate position (s), initial velocity (u), final velocity (v), acceleration (a), and time (t):
- v = u + at (velocity after time t)
- s = ut + ½at² (displacement after time t)
- v² = u² + 2as (velocity after displacement s)
- s = (u+v)/2 × t (displacement with average velocity)
These equations are extraordinarily powerful. They predict where a projectile will land, when a car will stop, or what velocity an object reaches. The key restriction: they only work for constant acceleration.
Graphs of Motion
A position-time graph shows how position changes with time. Its slope is velocity. A steep slope means high velocity; a flat line means zero velocity; a curved line means changing velocity (acceleration).
A velocity-time graph shows how velocity changes with time. Its slope is acceleration. The area under the curve represents displacement.
These graphs make the abstract concrete. Looking at them, you immediately see whether an object is speeding up, slowing down, or maintaining constant velocity.
Relative Velocity
If a boat travels at 10 m/s relative to water, and the water flows at 3 m/s relative to the shore, the boat's velocity relative to the shore is 10 + 3 = 13 m/s (if going with the current) or 10 - 3 = 7 m/s (if going against it). This is relative velocity—the velocity of one object measured from another object's perspective.
Relative velocity is vector addition. The key insight: motion always appears different from different reference frames. There's no absolute, universal "velocity"—only velocity relative to something else.
Applications
Understanding straight-line motion lets us solve real problems: How long does a car take to stop? How far does a stone fall in 3 seconds? When will two trains collide? These questions are answered using kinematic equations and careful attention to sign conventions.
Socratic Questions
- If a car's speedometer reads 60 km/h, does that tell you the car's velocity? What additional information do you need?
- Can an object have zero velocity but non-zero acceleration? Give a real-world example and explain why this seems counterintuitive.
- Why do the kinematic equations only work for constant acceleration? What would you need to do if acceleration were changing?
- Sketch a position-time graph for an object that first moves forward fast, then slows down, stops, and backs up slowly. What does the slope of this curve represent?
- If you're in a train moving at 50 m/s and throw a ball forward at 10 m/s (relative to the train), what is the ball's velocity relative to the ground?