Points A and B move along two adjacent
parallel paths. Initially A is 100 ft to the left of
B, and each point has a velocity of 5 fps to the
right. the velocity of A remains constant and the
acceleration of B is 2 fps2 to the left.
Determine the total distance traveled by B when
the two points pass each other.
Write equations for the positions of A and B vs time t.
Set them equal and solve for t. Point B may spend some time going right and then, after reversing direction, additional time going left. If that happens, you will need to calculate when A reverses direction and add the right- and left-going distances.
soo if we look at A as the orgin. The equation for X(A) is 5t and the Equation for x(B)= -2t^2+5t-100??
B does spend time going to the right before changing direction
To find the total distance traveled by point B when it passes point A, we need to first determine the time it takes for point A and point B to meet.
Let's denote the initial position of point A as A0 = -100 ft (to the left of point B), the velocity of point A as VA = 5 fps (to the right), and the initial position of point B as B0 = 0 ft (origin).
We can determine the position of point A as a function of time (t) using the formula:
PA = A0 + VA * t
Similarly, the position of point B as a function of time can be determined using the formula:
PB = B0 + VB * t + (1/2) * AB * t^2
Given that point B has an acceleration of AB = -2 fps^2 to the left, and its velocity is to the right with an initial velocity of VB0 = 0 ft/s, we can determine its velocity as a function of time as follows:
VB = VB0 + AB * t
For both points A and B to meet, their positions must be equal. So we can set up the equation:
PA = PB
Substituting the expressions for PA and PB:
A0 + VA * t = B0 + VB * t + (1/2) * AB * t^2
Substituting the values for A0, VA, B0, VB0, and AB:
-100 + 5t = 0 + (0 + (-2t))(t) + (1/2)(-2)(t^2)
Simplifying the equation:
-100 + 5t = -2t^2
Rearranging the terms to form a quadratic equation:
2t^2 + 5t - 100 = 0
We can solve this quadratic equation using factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
Substituting the values for a, b, and c:
t = (-(5) ± √((5)^2 - 4(2)(-100))) / (2(2))
Simplifying:
t = (-5 ± √(25 + 800)) / 4
t = (-5 ± √825) / 4
Since time cannot be negative in this context, we discard the negative solution:
t = (-5 + √825) / 4
Approximating the value of t to two decimal places:
t ≈ 4.27 seconds
Now that we have the time it takes for point A and point B to meet, we can calculate the distance traveled by point B during this time.
The distance traveled by point B is given by the equation:
Distance = VB0 * t + (1/2) * AB * t^2
Substituting the values for VB0, AB, and t:
Distance = 0 * 4.27 + (1/2) * (-2) * (4.27)^2
Simplifying the equation:
Distance = 0 + (1/2) * (-2) * 18.22
Calculating the value:
Distance = -18.22 ft
The total distance traveled by B when the two points pass each other is approximately 18.22 feet.