Hey- i have problems getting the answer for the last part of this problem- i've tried tons of different things and it is not as simple as just s(12)-(0)--- any help would be much appreciated!
A particle moves along a straight line and its position at time is given by s(t)=2t^3-18t^2+48t where s is measured in feet and t in seconds.
Find the velocity (in ft/sec) of the particle at time t=0 : s(t)=48
The particle stops moving (i.e. is in a rest) twice, at points: t= 4 and t= 2
What is the position of the particle at time 12? =1440
Finally, what is the TOTAL distance the particle travels between time 0 and time 12? ***???*****
A particle moves along a straight line and its position at time is given by s(t)=2t^3-18t^2+48t where s is measured in feet and t in seconds.
Find the velocity (in ft/sec) of the particle at time t=0 : s(t)=48
The particle stops moving (i.e. is in a rest) twice, at points: t= 4 and t= 2
What is the position of the particle at time 12? =1440
Finally, what is the TOTAL distance the particle travels between time 0 and time 12? ***???*****
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The answer is 1456
Explanation:
The particle moves in a one-dimensional space: along a line.
Its speed is s'(t)=6t^2-36t+48 = 8 (t-2)(t-4), as you computed.
This speed is positive for t in [0,2), then negative for t in (2,4), then positive for t in (4,12]
This means that the particle moves *forward* until t=2, then backward until t=4, then forward again.
-> At t=0, its position is 0.
-> At t=2, its position is 2*8 - 18*4 + 48*2 = 16 - 72 + 96 = 40 > 0
-> At t=4, its position is 2*64 - 18*16 + 48*4 = 32
-> at t=12, its position is 1440.
The movement of the particle being as described above, the total distance is: distance from t=0 to t=2 + distance from t=2 to t=4 + distance from t=4 to t=12.
This is |40-0| + |32-40| + |1440-32|, which is 40 + 8 + 1408 = 1456
Explanation:
The particle moves in a one-dimensional space: along a line.
Its speed is s'(t)=6t^2-36t+48 = 8 (t-2)(t-4), as you computed.
This speed is positive for t in [0,2), then negative for t in (2,4), then positive for t in (4,12]
This means that the particle moves *forward* until t=2, then backward until t=4, then forward again.
-> At t=0, its position is 0.
-> At t=2, its position is 2*8 - 18*4 + 48*2 = 16 - 72 + 96 = 40 > 0
-> At t=4, its position is 2*64 - 18*16 + 48*4 = 32
-> at t=12, its position is 1440.
The movement of the particle being as described above, the total distance is: distance from t=0 to t=2 + distance from t=2 to t=4 + distance from t=4 to t=12.
This is |40-0| + |32-40| + |1440-32|, which is 40 + 8 + 1408 = 1456