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Stopping distance and the arch of a bridge

Science and society, grade 12 (3 units) Lesson 6 of 14Learning

Stopping distance and the arch of a bridge

The braking distance

The braking distance is the way a car goes from the moment the driver presses the brake until it stands still. On a dry road, in metres, when the car drives at v\displaystyle v km/h:

d=0.005v2d=0.005v^{2}
v\displaystyle v (km/h)25\displaystyle 2550\displaystyle 50100\displaystyle 100
d\displaystyle d (metres)3.125\displaystyle 3.12512.5\displaystyle 12.550\displaystyle 50
  • Twice the speed gives 4 times the braking distance: (2v)2=4v2\displaystyle {(2v)^{2}}={4v^{2}}. From 50 to 100 km/h the distance grows from 12.5 to 50 metres.
  • The reason: a moving car’s energy grows as the square of its speed, and the brakes must take all of it away.
The reaction distance and the stopping distance

Before the driver presses the brake, about a second goes by: the driver sees the danger and moves a foot. Meanwhile the car goes on at the same speed, and covers about 0.3v\displaystyle 0.3v metres – the reaction distance.

The stopping distance is both together:

s=0.005v2+0.3vs=0.005v^{2}+0.3v
v\displaystyle v (km/h)30\displaystyle 3050\displaystyle 5080\displaystyle 80100\displaystyle 100
Reaction distance9\displaystyle 915\displaystyle 1524\displaystyle 2430\displaystyle 30
Braking distance4.5\displaystyle 4.512.5\displaystyle 12.532\displaystyle 3250\displaystyle 50
Stopping distance13.5\displaystyle 13.527.5\displaystyle 27.556\displaystyle 5680\displaystyle 80

On the board, x\displaystyle x is the speed; the solid line is the braking distance, and the dashed one the stopping distance:

y=0.005x2\displaystyle y=0.005x^{2}
y=0.005x2+0.3x\displaystyle y=0.005x^{2}+0.3x

At a low speed most of the stopping distance is the reaction distance, and at a high speed the braking distance.

The speed that lets a driver stop in time

A driver sees an obstacle 56 metres ahead. At what speed can the driver still stop before it?

0.005v2+0.3v=560.005v^{2}+0.3v=56
  • We multiply by 200: v2+60v=11200\displaystyle {v^{2}+60v}={11200}, that is v2+60v−11200=0\displaystyle {v^{2}+60v-11200}={0}.
  • By the quadratic formula: v=−60±3600+448002=−60±2202\displaystyle {v}={\frac{-60\pm\sqrt{3600+44800}}{2}}={\frac{-60\pm220}{2}}.
  • v=80\displaystyle {v}={80} or v=−140\displaystyle {v}={-140}.

A negative speed has no meaning: up to 80 km/h the driver stops before the obstacle.

A bridge’s arch

A bridge is held up by an arch shaped as a parabola.
The arch’s height above the water, in metres, x\displaystyle x metres from one of its ends: y=−0.02x2+2x\displaystyle {y}={-0.02x^{2}+2x}

y=−0.02x2+2x\displaystyle y=-0.02x^{2}+2x
y=32\displaystyle y=32
V=(50,50)\displaystyle V=(50,50)
A=(20,32)\displaystyle A=(20,32)
B=(80,32)\displaystyle B=(80,32)
  • The width: the arch leaves the water at x=0\displaystyle {x}={0} and comes back to it at x=100\displaystyle {x}={100} – the zeros. It is 100 metres wide.
  • The height: in the middle, at x=50\displaystyle {x}={50}, y=−0.02⋅2500+100=50\displaystyle {y}={-0.02\cdot2500+100}={50} metres.
  • A ship 32 metres high: we solve −0.02x2+2x=32\displaystyle {-0.02x^{2}+2x}={32}, and get x=20\displaystyle {x}={20} or x=80\displaystyle {x}={80}. The arch is higher than 32 metres only between 20 and 80 – a strip 60 metres wide, where the ship can pass.
Exercise 1

The braking distance on a dry road, in metres: d=0.005v2\displaystyle {d}={0.005v^{2}}

a

Work out the braking distance at 30 km/h.

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b

    Work out the braking distance at 60 km/h.

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      c

      Work out the braking distance at 120 km/h.

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        d

        How many times larger does the braking distance get when the speed grows from 30 to 60 km/h? And from 60 to 120? Explain.

        From 30 to 60

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          From 60 to 120

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            Exercise 2

            Near a school a child steps into the road 27.5 metres ahead of a car.
            The stopping distance, in metres: s=0.005v2+0.3v\displaystyle {s}={0.005v^{2}+0.3v}
            What is the highest speed at which the car can still stop before the child?

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                Exercise 3

                The entrance of a tunnel is shaped as a parabola.
                The ceiling’s height, in metres, x\displaystyle x metres from one side of the entrance: y=−0.125x2+2x\displaystyle {y}={-0.125x^{2}+2x}
                Mark the highest point and the zeros on the board, and check.
                How wide is the tunnel at the bottom, and how high is it in the middle?

                y=−0.125x2+2x\displaystyle y=-0.125x^{2}+2x

                The width at the bottom

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                  The height in the middle

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                    Exercise 4

                    A lorry 6 metres high drives into the tunnel of Exercise 3.
                    Where is the ceiling 6 metres high? How wide is the room the lorry has to pass?

                      The smaller x\displaystyle x

                        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                        The larger x\displaystyle x

                          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                          The width

                            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power