Calc1231W13T3 - Distance and speed

Sunday, 21 October 2018

6:38 PM

Untitled picture.png Brandishing your mathematical skills, you find a summer job helping out with the 2019 New Years Eve fireworks display on Sydney Harbour' 
503m 
The Sydney Harbour Bridge 
The lower arch of the Sydney Harbour Bridge is 503 metres in width and 118 metres high. It is also a parabola. If the zeros of the parabola are at x 
0 and x 
503, then the vertex is at 
the point [251.5,118] 
O _ Hence the equation of the parabola is 
y 
We will use this equation in the next few questions. 
Note: the Maple syntax for the point [1, 2] is (1, 2] _ 

Untitled picture.png Suppose a flaming ball is set to fire from [0,0] and fixed to the rail of the lower arch of the Sydney Harbour Bridge (ie the trajectory ofthe ball follows the rail). The ball must reach the point 
[503, 0] in exactly 6 seconds The x and y co-ordinates can be specified as a function oft in seconds, as 
• x(t) — 
. y(t) 
503/6*t 
118 
t(t — 6). 
9 
dc 
Note: We are assuming a constant x-velocity, ie 
should be constant. 
The distance travelled by the ball at time u seconds is given by the formula 
Hence after 2 seconds the ball has travelled (to the nearest metre) 
199 
Note: you will probably want to use Maple of WolframAlpha to calculate this integral. 
2 
dt. 
O 
_4_ 
metres_ 
Untitled picture.png 503 
6 
118 
9 
int( sqrt(diflx 
round ( %) 
-6) 
-6) 
472 
503 
6 
118 
9 
difly: t)2) 
475793 
4 
253009 
ln(503) — 
944 
253009 
475793 ) 
36 
199 
2499865 
253009 
ln(3) — 
253009 
ln(472 
2499865 ) 
Untitled picture.png 503 
6 
118 
9 
int( sqrt(diflx 
round ( %) 
-6) 
-6) 
472 
503 
6 
118 
9 
difly: t)2) 
475793 
4 
253009 
ln(503) — 
944 
253009 
475793 ) 
36 
199 
2499865 
253009 
ln(3) — 
253009 
ln(472 
2499865 )
Untitled picture.png Suppose a flaming ball is set to fire from [0,0] and fixed to the rail of the lower arch of the Sydney Harbour Bridge (ie the trajectory ofthe ball follows the rail). The ball must reach the point 
[503, 0] in exactly 6 seconds The x and y co-ordinates can be specified as a function oft in seconds, as 
• x(t) — 
. y(t) 
503/6*t 
118 
t(t — 6). 
9 
dc 
Note: We are assuming a constant x-velocity, ie 
should be constant. 
The distance travelled by the ball at time u seconds is given by the formula 
Hence after 2 seconds the ball has travelled (to the nearest metre) 
199 
Note: you will probably want to use Maple of WolframAlpha to calculate this integral. 
2 
dt. 
O 
_4_ 
metres_ 

Untitled picture.png The speed of the flaming ball is the rate of change in distance travelled_ Hence the speed of the flare at time u seconds is: 
d 
• After 2 seconds the ball is moving ata speed of 
88 
O metres per second (to the nearest integer). 
• The initial speed of the ball is 115 
O 
metres per second (to the nearest integer) 

Untitled picture.png 472 
Recall: that the equation describing the lower arch of the bridge is y 
253009 
The average height above launch level of the ball is (to the nearest metre) 79 
+ 118. 
O metres. 

Untitled picture.png Recall the formula for the speed at time u seconds is given by: 
du 
To accommodate even more fireworks in the 2019 schedule, the artistic director requires that the ball must now take only 3 seconds to exactly follow the trajectory of the lower arch. Hence 
the new initial speed is (to the nearest integer): 
230 
O metres per second 
Untitled picture.png 503 
6 
118 
9 
int( sqrt(diflx 
round ( %) 
-6) 
-6) 
472 
503 
6 
118 
9 
difly: t)2) 
475793 
4 
253009 
ln(503) — 
944 
253009 
475793 ) 
36 
199 
2499865 
253009 
ln(3) — 
253009 
ln(472 
2499865 ) 
Untitled picture.png = 2, sqrt(djfis 
= O, sqrt(dtfis 
diff y, t)2))) 
88 
diff y, t)2))) 
115 
Untitled picture.png М -472/253009 
503 
round( %) 
78_66666666 
79 
Untitled picture.png 503 
3 
subs(x = O, %) 
503 
944 
x + 0.9383697813 
253009 
229.9258334 
Untitled picture.png 503 
6 
118 
9 
int( sqrt(diflx 
round ( %) 
-6) 
-6) 
472 
503 
6 
118 
9 
difly: t)2) 
475793 
4 
253009 
ln(503) — 
944 
253009 
475793 ) 
36 
199 
2499865 
253009 
ln(3) — 
253009 
ln(472 
2499865 ) 
Untitled picture.png 503 
3 
subs(x = O, %) 
503 
944 
x + 0.9383697813 
253009 
229.9258334

 

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