Motion in a Straight Line

This chapter covers position, displacement, distance, speed, velocity, acceleration, graphs, uniformly accelerated motion, and relative velocity in one dimension.

Chapter overview

Motion in a straight line is one-dimensional motion. The position of a particle can be described using one coordinate, usually $x$.

1. Position, path length, and displacement

Position:

$$x=x(t)$$

Displacement:

$$\Delta x=x_2-x_1$$

Distance is the total path length travelled.

Important:

$$|\text{displacement}|\le\text{distance}$$

2. Average speed and average velocity

Average speed:

$$v_{\text{avg speed}} = \frac{\text{total distance}} {\text{total time}}$$

Average velocity:

$$v_{\text{avg}} = \frac{\Delta x}{\Delta t} = \frac{x_2-x_1}{t_2-t_1}$$

3. Instantaneous velocity

Instantaneous velocity is velocity at an instant.

$$v = \lim_{\Delta t\to0} \frac{\Delta x}{\Delta t}$$

In calculus notation:

$$v=\frac{dx}{dt}$$

4. Acceleration

Average acceleration:

$$a_{\text{avg}} = \frac{\Delta v}{\Delta t}$$

Instantaneous acceleration:

$$a = \frac{dv}{dt}$$

Also:

$$a=\frac{d^2x}{dt^2}$$

5. Uniform motion

If velocity is constant:

$$x=x_0+vt$$

For uniform motion:

$$a=0$$

6. Uniformly accelerated motion

If acceleration is constant, the equations of motion are:

$$v=u+at$$ $$s=ut+\frac{1}{2}at^2$$ $$v^2=u^2+2as$$ $$s=\frac{u+v}{2}t$$

7. Derivation of equations of motion

From acceleration:

$$a=\frac{v-u}{t}$$

Therefore:

$$v=u+at$$

Average velocity for constant acceleration:

$$v_{\text{avg}}=\frac{u+v}{2}$$

Displacement:

$$s=v_{\text{avg}}t$$ $$s=\frac{u+v}{2}t$$

Using $v=u+at$:

$$s=ut+\frac{1}{2}at^2$$

8. Graphs of motion

Position-time graph

Slope gives velocity.

$$v=\frac{\Delta x}{\Delta t}$$

Velocity-time graph

Slope gives acceleration.

$$a=\frac{\Delta v}{\Delta t}$$

Area under velocity-time graph gives displacement.

$$s=\int v\,dt$$

For constant acceleration:

$$s=\text{area of trapezium} = \frac{1}{2}(u+v)t$$

9. Motion under gravity

For freely falling body:

$$a=g$$

If upward direction is positive:

$$a=-g$$

Equations:

$$v=u-gt$$ $$h=ut-\frac{1}{2}gt^2$$ $$v^2=u^2-2gh$$

Maximum height:

$$H=\frac{u^2}{2g}$$

Time of ascent:

$$t=\frac{u}{g}$$

10. Relative velocity in one dimension

Velocity of $A$ relative to $B$:

$$v_{AB}=v_A-v_B$$

If two objects move in the same direction:

$$v_{\text{relative}}=|v_1-v_2|$$

If they move in opposite directions:

$$v_{\text{relative}}=|v_1+v_2|$$

11. Sign convention

12. Solved examples

Example 1

A car starts from rest and accelerates at $2\ \text{m s}^{-2}$ for $5$ s. Find velocity.

$$u=0,\quad a=2,\quad t=5$$ $$v=u+at=0+2(5)=10\ \text{m s}^{-1}$$

Example 2

Find distance covered.

$$s=ut+\frac{1}{2}at^2$$ $$s=0+\frac{1}{2}(2)(5)^2$$ $$s=25\ \text{m}$$

13. Common mistakes

• Confusing distance and displacement.
• Confusing speed and velocity.
• Forgetting signs in motion under gravity.
• Using equations of uniformly accelerated motion when acceleration is not constant.
• Forgetting area under $v-t$ graph gives displacement, not acceleration.

14. Quick revision