Velocity-Time Graphs

Velocity-Time Graphs – Key things to remember:

With speed on the $y$-axis and time on the $x$-axis, a speed-time graph tells us how someone/something’s speed has changed over a period of time.

1) The gradient of the line = Acceleration
2) Negative gradient = Deceleration
3) Flat section means constant velocity (NOT STOPPED)
4) Area under the graph = Distance travelled

Skill 1: Describing a graph

One Skill you will need learn is describing a velocity time graph.

Example: The speed-time graph shows a $50$-second car journey. Describe the $50$ second journey.

Step 1: Split the graph up into distinct sections, these can be seen in the image as $A, B, C$ and $D$.

Step 2: In detail describe each part of the journey, ensuring to use numerical values throughout.

Section $A$ – The car accelerated from $0$ to $15$ m/s over the first $10$ seconds (because the line is straight, the acceleration is constant).

Section $B$ –  The line is flat, meaning the car’s speed did not change for $10$ seconds – meaning it was moving at a constant speed.

Section $C$ – The car accelerated up to $25$ m/s over the next $10$ seconds,

Section $D$ –  Finally it spent the last $20$ seconds decelerating back down to $0$ m/s.

Skill 2: Calculating Acceleration

Acceleration is calculated as the change in speed over time.

Section $bf{A}$:  Acceleration between $0$s and $10$s $=$ gradient$=dfrac{15-0}{10-0}=1.5$ m/s$^2$

Section $bf{B}$: This section is flat, meaning the acceleration will be $0$

Section $bf{C}$: Acceleration between $20$s and $30$s $=$ gradient $=dfrac{25-15}{30-20}=1$ m/s$^2$

Section $bf{D}$: Acceleration between $30$s and $50$s $=$ gradient $=dfrac{0-25}{50-30}=dfrac{-25}{20} = -1.25$ m/s$^2$

Section $bf{A}$ has the largest acceleration, so the maximum acceleration is $1.5$ m/s$^2$

Note: units of acceleration are expressed in distance/time$bold{^2}$, which in this case is m/s$bold{^2}$.

Skill 3: Calculating total distance travelled

Calculating the total distance travelled is one of the most common exam questions you may see.

This gives two triangles, a rectangle, and a trapezium, which are all shapes that we can work out the area of.

$text{A}=dfrac{1}{2}times10 times15= 75$ m

$text{B}=10times15=150$ m

$C=dfrac{1}{2}(15+25)times10=200$ m

$D=dfrac{1}{2}times20times25=250$ m

Total distance travelled:

$75+150+200+250=675$ m

Skill 4: Average of curved graphs

To work out the average acceleration over the $4$ seconds, we will draw a line from where the graph is at $0$ s to where the graph is at $4$ s and find the gradient of it.

So, we get the average acceleration to be,

$text{gradient}=dfrac{6-0}{4-0}=1.5$ m/s$^2$

Skill 5: Instantaneous gradient of a curve

Then, we get the instantaneous acceleration to be,

$text{gradient}=dfrac{5.8-3.2}{3.5-1.0} = 1.04$ m/s$^2$ ($3$ sf).