62 Accelerated Motion

Acceleration

After the becomes large enough to balance out a skydiver’s , they will have no . From we already know that an object’s prevents a change in unless it experience a net force, so from that point when the forces are balanced and onward, the skydiver continues at a until they open their parachute.

During the initial part of skydive, before the is large enough to balance out the weight,  there is a so their changes.  The rate at which the velocity changes is known as the . 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 (a) during a certain time interval (Δt) by subtracting the (vi) from the  (vf) to get the change in velocity (Δv) and then dividing by 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 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 are m/s/s or equivalently m/s2

Acceleration Direction

The direction of depends on the direction of the change in . 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 direction and change and the resulting 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

A car is moving in the positive direction and slams on the brakes. What direction is the ?

A car is moving in the negative direction and slams on the brakes. What direction is the acceleration?

A car is moving in the negative direction and speeding up. What direction is the acceleration?

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