Unit 23: Applied mathematics for engineering
LO2, 3, 4, 5
Roller Coaster
Learner activity sheet
You are required to read the following description and complete the questions presented in Activities 1
and 2.
A roller coaster track comprises three connected sections with continuous gradients at the joins. The first
section of track starts at ground level and continues upwards with a constant gradient until it reaches a
point where the horizontal distance from the start is 15 m and the height above ground level is 15 m. The
track then continues along a second section whose shape is given by the formula:
h2  ax2  bx  c
where
a, b and c are constants,
h2 m is the height above ground level,
x m is the horizontal distance measured from the start of the first section.
The second section continues until the total horizontal distance from the start of the track is 30 m and the
height is 12 m. The third section of track then continues whose shape is given by the formula:
h3  dx3  ex 2  fx  g
where
d, e, f and g are constants,
h3 m is the height above ground level,
x m is the horizontal distance measured from the start of the first section.
This third section of track has a continuous, diminishing, downward gradient until it reaches ground level
where the total horizontal distance from the start of the track is 50 m and the gradient is zero.
Version 1
Activity 1
(i)
From the information given, sketch a graph of the complete roller coaster track showing height
against horizontal distance.
(ii) Formulate and solve linear simultaneous equations to calculate the values a, b, c, d, e, f and g.
(iii) Calculate the highest point of the track.
(iv) Use a spreadsheet to produce a graph of the track with the values of a, b, c, d, e, f and g calculated in
part (ii) and confirm that your sketch in part (i) is reasonable.
Activity 2
A carriage together with its passengers has a total mass of 1000 kg. The carriage has a speed of 4 ms–1 at
the highest point on the track within the second section when it starts to roll down towards the third
section. Assume that the train is modelled by a point mass moving in the single vertical plane and that
resistance to motion due to friction and aerodynamic drag can be neglected.
(i)
Calculate the potential energy of the carriage, relative to ground level, as it starts its decent from the
highest point on the track.
(ii) Calculate the kinetic energy of the carriage as it starts its decent.
(iii) Calculate the speed of the carriage as it passes the point on the track where the second and third
sections meet.
(iv) Calculate the final speed of the carriage at the end of the track where it reaches ground level.
Version 1