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.
To determine the total distance traveled by point B when the two points pass each other, we first need to find the time it takes for them to meet.
Let's assume that B travels a distance of "d" feet before meeting point A. Since both A and B have a velocity of 5 fps to the right, B takes d/5 seconds to cover that distance.
Now, let's determine the time it takes for point B to meet point A. Since A is initially 100 ft to the left of B, they need to cover a total distance of 100 ft to meet each other. Since A's velocity remains constant at 5 fps and B has an acceleration of 2 fps^2 to the left, we can use the formula:
Distance = Initial Velocity * Time + (1/2) * Acceleration * Time^2
Substituting the known values:
100 ft = 5 fps * Time + (1/2) * (-2 fps^2) * Time^2
Simplifying the equation:
100 ft = 5 ft/s * Time - 1 ft/s^2 * Time^2
Rearranging the equation to the standard quadratic form:
0 = -1 ft/s^2 * Time^2 + 5 ft/s * Time - 100 ft
We can solve this quadratic equation using the quadratic formula:
Time = [-b ± √(b^2 - 4ac)] / 2a
Applying the values:
Time = [-5 ± √(5^2 - 4 * (-1) * (-100))] / (2 * (-1))
Time = [-5 ± √(25 + 400)] / (-2)
Time = [-5 ± √425] / (-2)
Time ≈ -4.78 s or Time ≈ 8.78 s
Since time can't be negative in this scenario, we discard the negative value and keep the positive one. Thus, it takes approximately 8.78 seconds for point A and point B to meet.
Now that we know the time, we can determine how far point B has traveled. Using the velocity formula:
Distance = Initial Velocity * Time + (1/2) * Acceleration * Time^2
Substituting the known values:
Distance = 5 ft/s * 8.78 s + (1/2) * (-2 ft/s^2) * (8.78 s)^2
Simplifying the equation:
Distance = 44 ft/s * 8.78 s + (-1 ft/s^2) * 4.39 s^2
Distance = 385.12 ft - 38.512 ft
Distance ≈ 346.608 ft
Therefore, the total distance traveled by point B when the two points pass each other is approximately 346.608 feet.