Motion in a Straight Line — Study Notes

1. Introduction

Every object in the universe is in motion. To describe motion precisely we must fix a reference frame - a coordinate system relative to which the position of the object is specified. Motion along a straight line, also called rectilinear motion, is the simplest kind of motion and is the natural starting point for the study of mechanics. In this chapter we describe the position of an object, its path length and displacement, and its velocity and acceleration, all along a single axis.

We use the concepts of average and instantaneous values to deal with motion that is not steady. We then derive the equations of uniformly accelerated motion and apply them to objects falling freely under gravity. Finally, we study the concept of relative velocity, which allows us to understand motion as seen from different moving observers.

The entire analysis in this chapter treats the moving object as a particle - a point object whose internal structure does not matter. This approximation greatly simplifies the mathematics and is valid whenever the size of the object is small compared with the distances over which it moves.

2. Position, Path Length and Displacement

The position of an object along a straight line is specified by its coordinate x on an axis, measured from a chosen origin. If the object is at x1 at time t1 and at x2 at time t2, then we can define two different quantities to describe the change: path length and displacement.

The path length is the total distance actually covered by the object during its motion. It is a scalar quantity, always positive, and its SI unit is the metre. The displacement is the change in position, defined as the difference between the final and initial positions:

delta x = x2 - x1

Displacement is a vector quantity. It can be positive, negative, or zero. The magnitude of displacement is the shortest distance between the initial and final positions. If an object moves from x = 10 m to x = 2 m, its displacement is -8 m, while its path length is 8 m. If the object then returns to x = 10 m, the path length is 16 m but the total displacement is zero.

3. Average Velocity and Average Speed

The average velocity is the displacement divided by the time interval:

v_avg = delta x / delta t = (x2 - x1) / (t2 - t1)

The average speed is the total path length travelled divided by the total time taken. Average speed is a scalar, while average velocity is a vector. In general the average speed is greater than or equal to the magnitude of the average velocity, and they are equal only when the motion is in a single direction without reversal.

If the velocity changes uniformly, the average velocity is the arithmetic mean of the initial and final velocities:

v_avg = (u + v) / 2

where u is the initial velocity and v the final velocity.

4. Instantaneous Velocity and Acceleration

The instantaneous velocity is the velocity at a particular instant of time. It is defined as the limit of the average velocity as the time interval approaches zero:

v = limit (delta t -> 0) of delta x / delta t = dx/dt

The instantaneous velocity equals the slope of the position-time graph at that instant. Similarly, acceleration is the rate of change of velocity:

a = dv/dt

Acceleration is a vector quantity with SI unit m/s^2. When the velocity decreases, the acceleration is negative and is called retardation or deceleration. Since velocity is the derivative of position and acceleration is the derivative of velocity, the position is obtained by integration:

x(t) = integral of v(t) dt and v(t) = integral of a(t) dt

5. Kinematic Equations for Uniformly Accelerated Motion

For motion with constant acceleration a, initial velocity u, and final velocity v after time t, the following equations hold:

v = u + at

s = ut + (1/2)at^2

v^2 = u^2 + 2as

Here s is the displacement. These are the three equations of motion. They are valid only when the acceleration is constant. A useful fourth relation gives the displacement in the nth second:

s_n = u + a(n - 1/2)

The position-time graph for uniformly accelerated motion is a parabola, while the velocity-time graph is a straight line with slope equal to the acceleration.

6. Free Fall

Motion under gravity is the most important example of uniformly accelerated motion. When a body falls freely near the surface of the Earth, its acceleration is the acceleration due to gravity g, whose standard value is about 9.8 m/s^2 (often taken as 10 m/s^2 for convenience). The equations of motion apply with a = g.

For a body dropped from rest, the velocity after time t is v = gt and the distance fallen is h = (1/2)gt^2. For a body thrown vertically upward with initial velocity u, the maximum height reached is:

h_max = u^2 / (2g)

The time taken to reach the maximum height is t = u/g, and the total time of flight is 2u/g. An important fact is that the speed with which the body returns to the ground equals the speed with which it was thrown up.

7. Relative Velocity

The relative velocity of object A with respect to object B is the velocity of A as seen by an observer moving with B. If both move along the same straight line with velocities vA and vB, then:

v_AB = vA - vB

If the two objects move in the same direction, their relative velocity is the difference of their speeds. If they move in opposite directions, the relative velocity is the sum of their speeds. For example, two trains each moving at 60 km/h in the same direction have zero relative velocity, while if they move toward each other, their relative velocity is 120 km/h.

Relative velocity explains how a person walking forward inside a moving train appears to a person standing on the platform: the velocities add, giving the resultant speed of the walker relative to the ground.

Quick Revision Tables

Mind Map

Important Diagrams (SVG)

Diagram 1: Velocity-Time Graph and Equations of Motion

Diagram 2: Projectile and Free Fall Motion under Gravity

Common Mistakes

1. Confusing path length with displacement; path length is total distance travelled while displacement is the change in position and can be zero.
2. Assuming average speed equals the magnitude of average velocity; they are equal only for motion in one direction without reversal.
3. Using the equations of motion when acceleration is not constant; they apply only to uniformly accelerated motion.
4. Forgetting that at the highest point of a vertical throw, the velocity is zero but acceleration is still g downward.
5. Adding velocities instead of taking the difference when computing relative velocity for objects moving in the same direction.
6. Ignoring the sign of displacement and acceleration, which are vector quantities.
7. Writing the displacement in the nth second formula incorrectly; it is s_n = u + a(n - 1/2).

Exam Tips

1. Define displacement, average velocity, instantaneous velocity, and acceleration with their SI units.
2. Write all three equations of motion and the condition under which they are valid (constant acceleration).
3. State that the slope of a velocity-time graph gives acceleration and the area under it gives displacement.
4. Solve free-fall problems with g = 9.8 m/s^2, remembering that upward is positive and downward is negative.
5. Use relative velocity v_AB = vA - vB and explain whether speeds add or subtract for the given directions.
6. Practise numericals on average velocity = displacement / time and average speed = path length / time.
7. Draw and label position-time and velocity-time graphs for uniform and uniformly accelerated motion.

Conclusion

In this chapter we learned to describe motion along a straight line using the concepts of position, path length, and displacement. We distinguished between average and instantaneous velocity and acceleration, and saw how they are related through differentiation and integration. The three equations of motion - v = u + at, s = ut + (1/2)at^2, and v^2 = u^2 + 2as - provide a complete description of uniformly accelerated motion, with free fall under gravity as the most important example. We also understood relative velocity, which lets us compare the motion of two objects as seen from each other. These tools of kinematics will now be extended to two dimensions in the study of motion in a plane.