The position vectors, r km, and velocity vectors, vkm/h, of two boats at a certain time are given by
Boat A: 10 am, r(a) = 6i+10j, v(a) = 8i-2j
Boat B: 10:30 am, r(b) = 8i+5j, v(b) = 4i+j
If the boats collide, find the time when they collide as well as the position vectors of the point of collision.
If the boats do not collide, determine the time at which Boat b should head out so that two boats collide?
To find the time when the two boats collide, we need to determine if and when their position vectors intersect.
First, let's find the parametric equations for the position vectors of the two boats. We'll use the time t as the parameter.
For Boat A:
r(a) = 6i + 10j
v(a) = 8i - 2j
The parametric equations for the position vector of Boat A are:
x(a) = 6t + 6
y(a) = 10t + 10
Similarly, for Boat B:
r(b) = 8i + 5j
v(b) = 4i + j
The parametric equations for the position vector of Boat B are:
x(b) = 8t + 8
y(b) = 5t + 5
To find the time when the boats collide, we need to solve the system of equations formed by the parametric equations of their position vectors:
6t + 6 = 8t + 8 (x-component equation)
10t + 10 = 5t + 5 (y-component equation)
Simplifying the equations:
-2t = 2 (x-component equation)
5t = -5 (y-component equation)
Solving for t:
t = -1 (x-component equation)
t = -1 (y-component equation)
Since both equations give us t = -1, the boats collide at t = -1. However, since negative time does not make sense in this context, we can conclude that the boats do not collide.
To determine the time at which Boat B should head out so that the boats collide, we need to find a time t where their position vectors intersect.
By equating the x-components:
6t + 6 = 8t + 8
Simplifying:
-2t = 2
Solving for t:
t = -1
The time t = -1 indicates when the two boats would meet if Boat B starts at its initial position. However, since negative time does not make sense, we need to add the time it would take for Boat B to reach its initial position.
Boat B started at 10:30 am. Let's assume it takes t0 hours for Boat B to reach its initial position.
So, the corrected time for Boat B to head out is:
10:30 am + t0 hours
Therefore, Boat B should head out at 10:30 am + t0 hours to collide with Boat A.
To find the time when the boats collide, we need to determine if there is a time t where the position vectors of the two boats are equal.
Let's assume that the boats collide at time t. At time t, the position vectors can be written as:
For Boat A:
r(a,t) = 6i + 10j + (8i - 2j)t
For Boat B:
r(b,t) = 8i + 5j + (4i + j)t
Setting these two position vectors equal to each other:
6i + 10j + (8i - 2j)t = 8i + 5j + (4i + j)t
Simplifying the equation:
14i + 8j + (6i -3j)t = 0
We can separate the x-component and y-component:
14i + (6i - 3j)t = 0
8j = 0
From the second equation, we can see that j = 0. This means that the boats must collide on the x-axis.
Substituting j = 0 into the first equation:
14i + (6i - 3(0))t = 0
Simplifying the equation:
20i = 3t
Therefore, t = (20/3) hours.
To find the position vectors at the point of collision, we substitute t = (20/3) into either of the original position vectors. Let's use Boat A's position vector:
r(a, t) = 6i + 10j + (8i - 2j)(20/3)
Simplifying the equation:
r(a, t) = 6i + 10j + (160/3)i - (40/3)j
Combining like terms:
r(a, t) = (6 + (160/3))i + (10 - (40/3))j
Simplifying further:
r(a, t) = (74/3)i + (30/3 - 40/3)j
r(a, t) = (74/3)i - (10/3)j
Therefore, at time t = (20/3) hours, the position vectors of the point of collision are r = (74/3)i - (10/3)j km.
If the boats do not collide, we need to find the time at which Boat B should head out so that the two boats collide. Since the boats must collide on the x-axis, the y-components of the position vectors should be equal.
Setting the y-components equal to each other:
10 + (-2)t = 5 + jt
Simplifying the equation:
5 + (-2 - j)t = 0
Setting the equation equal to zero:
-2 - j = 0
j = -2
Therefore, for the two boats to collide, Boat B should head out with a velocity vector of v(b) = 4i - 2j km/h.