13 Acceleration

The rate at which the velocity changes is known as the acceleration. Note that students often confuse velocity and acceleration because they are both rates of change, so to be specific: velocity defines the rate at which the position is changing and acceleration defines the rate at which the velocity is changing.  We can calculate the average acceleration (a) during a certain time interval (Δt) by subtracting the initial velocity (vi) from the final velocity (vf) to get the change in velocity (Δv) and then dividing by the time interval (Δt):

(1)   \begin{equation*} \bold{a} = \frac{\bold{\Delta v}}{\Delta t} =\frac{\bold{v_f-v_i}}{\Delta t} \end{equation*}

Everyday Example

Let’s calculate the average acceleration during the roughly 2 seconds it takes a parachute to fully open and slow a skydiver from 120 MPH to 6.0 MPH. First let’s remember that the skydiver is moving in our negative direction so the initial and final velocities should be negative. Also, lets convert to meters per second: \bold{v_f} = -6.0 \,\bold{mph} = -2.7 \,\bold{m/s} and \bold{v_i} = \bold{v_f} = -120 \,\bold{mph} -54 \,\bold{m/s}.

Starting with our definition of acceleration:

    \begin{equation*} \bold{a} =\frac{\bold{v_f-v_i}}{\Delta t} \end{equation*}

Inserting our values:

    \begin{equation*} \bold{a} =\frac{-2.7 \,\bold{m/s}-(-54 \,\bold{m/s})}{2\,\bold{s})} \end{equation*}

The two negatives in front of the 54 m/s make a positive, and then we calculate a value.

    \begin{equation*} \bold{a} =\frac{-2.7 \,\bold{m/s}+(54 \,\bold{m/s})}{2\,\bold{s)}} = 26 \,\bold{m/s/s} \end{equation*}

We now get a chance to see that the units of acceleration are m/s/s or equivalently m/s2

Acceleration Direction

The direction of acceleration depends on the direction of the change in velocity. If the velocity becomes more negative, then acceleration must be negative. This is the case for our skydiver during the first part of the jump; their speed is increasing in the negative direction, so their velocity is becoming more negative and therefore acceleration is negative. Conversely, if an object moves in the negative direction, but slows down, the acceleration is positive, even though the velocity is still negative! This was the case for our skydiver just after opening their parachute, when they still moved downward, but were slowing down. Slowing down in the negative direction means the velocity is becoming less negative, so the acceleration must be positive. All of the possible combinations of velocity direction and speed change and the resulting acceleration are summarized in the following chart:

Table Showing Possible Acceleration Directions
Initial direction of motion ( initial velocity direction) Speed change Direction of Acceleration
positive speeding up positive
positive slowing down negative
negative speeding up negative
negative slowing down positive

Reinforcement Exercises

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