The position vectors, r km, and the velocity vectors, v km/h, of two boats at a certain time are given by ...
Boat A:at 10:00 am rA=6i+10j vA=8i–2j
Boat B:at 10:30 am rB=8i+5j vB=4i+j
... determine the time at which boat B should head out so that the two boats do collide.
If A starts at time 0, and B starts at time k,
(6+8t)i + (10-2t)j = (8+4(t-k-1/2))i + (5+(t-k-1/2))j
so,
6+8t = 8+4(t-k-1/2)
10-2t = 5+t-k-1/2
(k,t) = (-11/8,11/8)
B had to leave at 10:00 - 1:122:30 = 8:37:30
At t=11/8 = 11:22:30, both boats are at (17,7.25)
To determine the time at which boat B should head out so that the two boats collide, we need to calculate their positions at different times and see when they coincide.
First, let's consider boat A's position after B has set out. Let's assume boat B sets out at time t hours. Therefore, at time t = 0, boat A will be at position rA = 6i + 10j.
Now, let's calculate the position of boat A at time t = 0.5 hours (10:00 am + 0.5 hours = 10:30 am). Using the velocity vector vA = 8i - 2j, we can calculate the displacement vector dA = vA * (t = 0.5).
dA = (8i - 2j) * (0.5) = 4i - j
Adding this displacement to the initial position of boat A, we get:
rA' = rA + dA = (6i + 10j) + (4i - j) = 10i + 9j
Now, let's calculate the position of boat B at time t = 0.5 hours using the same method. The initial position of boat B is rB = 8i + 5j, and the velocity vector is vB = 4i + j.
Using vB = 4i + j, we calculate the displacement vector dB = vB * (t = 0.5):
dB = (4i + j) * (0.5) = 2i + 0.5j
Adding this displacement to the initial position of boat B, we get:
rB' = rB + dB = (8i + 5j) + (2i + 0.5j) = 10i + 5.5j
The positions of the two boats at time t = 0.5 hours are rA' = 10i + 9j and rB' = 10i + 5.5j, respectively.
For the two boats to collide, their positions need to be the same. Therefore, we can equate the x and y components of their positions:
10i + 9j = 10i + 5.5j
By comparing the x-components and y-components separately, we get:
10 = 10
9 = 5.5
Since the y-components are not equal, the two boats will never collide. Therefore, there is no specific time at which boat B should head out to collide with boat A.
To determine the time at which the two boats collide, we need to find the time when their position vectors are equal.
The position vector of Boat A at a certain time t is given by rA = 6i + 10j, where i and j represent the unit vectors for the x and y directions respectively. The position vector of Boat B at a certain time t is given by rB = 8i + 5j.
We can set up two equations using the x and y components of the position vectors:
Equation 1: 6 + 8t = 8 (for the x component)
Equation 2: 10 + 5t = 5 (for the y component)
To solve for t, we can solve these two equations simultaneously.
Equation 1 simplifies to:
8t = 2
t = 2/8
t = 1/4
Equation 2 simplifies to:
5t = -5
t = -5/5
t = -1
Since time cannot be negative, we disregard the negative value of t. Therefore, the time at which Boat B should head out so that the two boats collide is t = 1/4 hour or 15 minutes.
So, Boat B should head out 15 minutes (or at 10:15 am) after Boat A.