Overtaking

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• Published : May 9, 2012

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| | |Year 12 Math C assignment | |Overtaking | | | | | | | | | | |

The problem

Aim: To find the maximum speed of Corolla KE5 SEDAN when following a delivery van travelling consistently at 55kmh.

Case 1.0 Corolla overtakes the van in a straight line motion. Based on the 1st 8m in the graph given.

Assumptions:
• Both Corolla and delivery van are driving in a straight line motion, which means no making U-turns or driving back-forward • Both Corolla and delivery van are moving at constant speed • Three-seconds rule is applied for the minimum safe distance between the two cars • The weather is in a good condition during daylight with good, dry roads • The Corolla KE5 SEDAN is a third generation model produced and sold in 1977-1979. The length of the Corolla is 3995mm(approx.4m) (wiki.okwave.jp/wikipedia/index/Toyota_corolla)  • The delivery van is assumed to be a Ford 3-door third generation van which is produced and sold between in 1975-1991. Its length is 5253mm(approx.5.2m) (en.wikipedia.org/wiki/Ford_Falcon)

The minimum safe distance between the two cars can be calculated as the speed of Ford Van and time (Three-second rule) are known.

[pic]
[pic]
[pic]
[pic]
[pic]

[pic]
It is assumed in this case that, the distance between the Corolla and the Ford van is 46m. [pic]

The diagram below represents the situation of the model, the distance between the two cars and the length of two cars. [pic]

According to the graph given, the corolla accelerates constantly until t =8. Its linear formula can be calculated as follows:
[pic]
[pic]
[pic]
[pic]
[pic]

[pic]
[pic]
[pic]
[pic]

Since it is a velocity equation, this can also be written as: v= 5t + 55
The numbers are in km/h. Therefore it is converted into m/s. (International System of Units) v=[pic]t + [pic]
v=1.39t + 15.28
[pic]The velocity for Corolla is v = 1.39t + 15.28
[pic]

Once the velocity is found, the displacement and the acceleration of corolla can also be found using calculus. The acceleration of corolla is found by differentiation.
V= 1.39t + 15.28
[pic]
[pic] = 1.39
[pic]
[pic]Corolla is accelerating at the speed of 1.39ms-2.

The displacement of the corolla is found by integration.
V=1.39t + 15.28
S= [pic]
S = [pic]
[pic]The displacement of Corolla is [pic]
However, when t=0, S=0.
[pic]C must equal zero.
[pic]The displacement of corolla is [pic]
[pic]
The information is given that the delivery van travels consistently at 55kmh. Therefore the velocity of the van is:
V= 55
Its number is in km/h. Therefore it is converted into m/s. (International System of Units) V = [pic]
V= 15.28
∴The velocity of the van is 15.28
Once the velocity is found, its acceleration and displacement can also be found using calculus. The acceleration of van is found by differentiation.
V=15.28
a= [pic]
a=0
∴The delivery van does not accelerate because it’s travelling at a constant speed of 15.28m/s.

The displacement of the van is found using integration.
V= 15.28
S= [pic]
S = 15.28t + C
[pic]The displacement of the van is 15.28t + C
However the van and corolla should have the same origin.
Therefore S = 15.28t + initial difference between the cars + Length of Corolla = 15.28t + 46 + 4
= 15.28t + 50
[pic]The displacement of...