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Motion in a Straight Line
Introduction
Instantaneous Velocity and Speed
Practice: Position, Velocity and Speed
Acceleration
Practice: Acceleration and Motion Graphs
Kinematic Equations for Uniformly Accelerated Motion
Practice: Equations of Motion
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Motion in a Straight LineInstantaneous Velocity and Speed

Reading time: ~20 min

Sina jogs from the school gate to a lamp post 120 m down the lane, rests there for 20 s, and jogs back. How fast did she go? There are two honest answers, and they are not the same.

The average velocity over an interval is the displacement divided by the time taken:

\bar{v} = \dfrac{x_2 - x_1}{t_2 - t_1} = \dfrac{\Delta x}{\Delta t}

The average speed is the total path length divided by the total time. On an x–t graph the average velocity is the slope of the chord joining the two ends of the interval. The blue graph below is Sina's jog, and the amber chord joins her positions at the times you choose.

Choose intervals that complete all three missions.

    Choose the interval: t_1 = ${t1} s and t_2 = ${t2} s.

    So the two averages answer different questions. Over the whole jog, Sina's displacement is zero, so her average velocity is zero, but she ran 240 m, so her average speed is not. You would not tell a tired runner that her average speed was zero! That is why average speed is defined with the path length. Since the path length is never less than the size of the displacement, the average speed is never less than the size of the average velocity; the two are equal only when the body does not turn back during the interval.

    From a Chord to a Tangent

    An average tells us how fast a body moved over an interval, but not how fast it was moving at one particular instant, which is what a speedometer shows. The idea is to shrink the interval. Take a car whose position is x = 0.08 t³ (x in metres, t in seconds), and the instant t = 4 s, point P. Take an interval Δt centred on P, from t_1 = 4 - \tfrac{\Delta t}{2} to t_2 = 4 + \tfrac{\Delta t}{2}, and work out the average velocity Δx/Δt over it.

    Pick each Δt in turn. Each one draws the amber chord through P₁ and P₂ and fills one row of Table 2.1.

    Fig. 2.1 — the x–t graph of the car, x = 0.08 t³, with point P at t = 4 s.

    Table 2.1 — the limiting value of Δx/Δt at t = 4 s

    As Δt shrinks, Δx/Δt settles on 3.84 m s⁻¹, and the amber chord swings onto the green tangent to the curve at P.

    That limit is the instantaneous velocity, or simply the velocity, at that instant:

    v = \displaystyle\lim_{\Delta t \to 0} \dfrac{\Delta x}{\Delta t} \qquad (2.1a)

    v = \dfrac{dx}{dt} \qquad (2.1b)

    In the language of calculus, dx/dt is the derivative of x with respect to t: the rate at which position changes at that instant. On a graph, the velocity at an instant is the slope of the tangent to the x–t graph at that instant. For our car, x = 0.08 t³ gives \dfrac{dx}{dt} = 0.24\,t^2, which at t = 4 s is m s⁻¹, exactly the limit in the table.

    Drawing tangents on a carefully plotted graph is slow and not very accurate. When we know the positions at many instants, we can shrink Δt numerically as in Table 2.1; when we know x as a formula in t, calculus gives the velocity at every instant at once, as in the next example.

    Velocity from a Formula

    Example 2.1 The position of an object moving along the x-axis is x = a + bt², where a = 8.5 m, b = 2.5 m s⁻² and t is in seconds. What is its velocity at t = 0 s and at t = 2.0 s? What is its average velocity between t = 2.0 s and t = 4.0 s?

    Solution: The velocity is

    v = \dfrac{dx}{dt} = \dfrac{d}{dt}\left(a + bt^2\right) = 2bt = m s⁻¹.

    At t = 0 s, v = m s⁻¹, and at t = 2.0 s, v = m s⁻¹.

    The average velocity between 2.0 s and 4.0 s is

    \dfrac{x(4.0) - x(2.0)}{4.0 - 2.0} = \dfrac{a + 16b - a - 4b}{2.0} = 6.0\,b

    which is m s⁻¹.

    x = 8.5 + 2.5 t². Each answer you give is drawn here.

    The green tangents have slopes 0 and 10 m s⁻¹, the velocities at those two instants, while the amber chord from t = 2 s to 4 s has slope 15 m s⁻¹, the average over that interval. For uniform motion there is nothing to choose between them: the x–t graph is a straight line, and the velocity at every instant equals the average velocity.

    Speed Is the Size of Velocity

    The instantaneous speed, or simply the speed, is the magnitude of the velocity. A velocity of +24.0 m s⁻¹ and a velocity of −24.0 m s⁻¹ both have a speed of 24.0 m s⁻¹.

    Over a finite interval, Sina's jog showed that the average speed can be bigger than the size of the average velocity. But at a single instant, the speed is always exactly equal to the size of the velocity. Why?

    Because speed and velocity are the same thing
    Because over a vanishingly short time the body cannot turn back, so the path length and the size of the displacement become equal
    Because the average speed always equals the size of the average velocity

    The two averages differ only because a body can turn back inside an interval. Shrink the interval to an instant and there is no time to turn back: path length and the size of the displacement shrink together, and so do the two averages built from them.

    Speed Round: Velocity

    For a formula, differentiate: v = dx/dt. For a graph, read the slope of the tangent. For an average, divide Δx by Δt.

    Get five right to move on.

    Sina