A car drives along a road 𝑃𝑄. The car starts at 𝑃 and takes 12
seconds to reach point 𝑄. The graph shows the horizontal
distance, 𝑑, of the car from point 𝑃 at a time of 𝑡 seconds after it starts its
journey. The car drives at a constant
Part a) Which of the following
diagrams represents the top-down view of the road 𝑃𝑄? Circle the letter.
This top-down view is what we would
see if we were in an airplane looking down at the road. The horizontal distance, 𝑑, tells
us how far the car is in the horizontal direction from point 𝑃.
We see that the car is travelling
away from 𝑃 at a consistent rate for about four seconds. After that, we see that the car is
still travelling away from 𝑃, but not quite as quickly as before. And then the horizontal distance
becomes even at 90 metres. From about eight seconds to 12
seconds, the car is no longer travelling away from point 𝑃 in a horizontal way.
If we’re looking from the top down
at our road, from left to right will be the horizontal distance. Let’s consider the four roads. On road A, the car travels away
from point 𝑃 throughout the whole road. If we graph this as a relationship
of the time the car is travelling and 𝑑, the horizontal distance away from 𝑃, it
would look like this. The longer the car drives, the
further away it gets from 𝑃 in a horizontal direction.
If we graph road B, the time the
car is travelling compared to how far away from 𝑃 it is, it would look like
this. The car is slowly moving away from
𝑃 and then more quickly moving away from 𝑃 in the horizontal direction and finally
driving away from 𝑃 in the horizontal direction very little.
Following our car again, we want to
relate the time that it travels to its horizontal distance. On this road, the car travels away
from 𝑃, turns, still travelling away from 𝑃, but not as quickly, and then stops
travelling horizontally away from 𝑃. Based on the graph we’re given, the
relationship between time and horizontal distance for a road 𝑃𝑄 would look like
If we wanted to consider road D,
the relationship would look like this. The car travels away from 𝑃
horizontally and then turns and then travels away from 𝑃 horizontally again. Road C is the best reflection of
our road 𝑃𝑄.
For part b), use the graph to
estimate the horizontal speed of the car six seconds after it begins its
We know that speed equals the
distance divided by the time. We’re interested in the speed of
this car at 𝑡 equals six. To calculate the speed at six
seconds, we need to consider the gradient of the line that intersects this
point. We’ll consider the way the
horizontal distance is changing in relation to the way that the time is changing:
the change in horizontal distance over the change in time.
The change in horizontal distance
has gone from 60 metres to 100 metres. That makes our numerator 100 metres
minus 60 metres. Our time has changed from two
seconds to nine seconds. The denominator will then be nine
seconds minus two seconds.
100 minus 60 equals 40, 40
metres. Nine minus two equals seven, seven
seconds. If we divide 40 by seven, we get
5.7142 continuing metres per second. Rounded to two decimal places, we
consider the digit to the right, which is the deciding digit. In this case, we have a four, which
is less than five. So we’ll round down.
By rounding to two decimal places,
we get 5.71 metres per second. An estimate horizontal speed at a
time of six seconds is 5.71 metres per second.