Final answer:
The particle's distance from the origin after 8 seconds can be calculated using kinematic equations, which results in a distance of approximately 284.3 meters.
Step-by-step explanation:
To calculate the particle's distance from the origin after 8 seconds, we will use the kinematic equations for motion under constant acceleration. The initial velocity vector ℓ is given by 2i - 4j + 3k and the acceleration vector a is given by -4i + 3j - 8k. After 8 seconds, the position vector ℓ(t) can be determined from:
ℓ(t) = ℓ0 + vt + ½at2
Where ℓ0 is the initial position, v is the initial velocity, a is the acceleration, and t is time. Since the particle starts at the origin, ℓ0 = 0. Plugging in the values:
ℓ(8s) = 0 + (2i - 4j + 3k)(8s) + ½(-4i + 3j - 8k)(8s)2
Simplifying gives:
ℓ(8s) = (16i - 32j + 24k) + ½(-4i + 3j - 8k)(64)
ℓ(8s) = (16i - 32j + 24k) - (128i - 96j + 256k)
ℓ(8s) = (-112i - 128j - 232k) meters
To find the distance from the origin, calculate the magnitude of the position vector:
Distance = |ℓ(8s)| = √((-112)2 + (-128)2 + (-232)2) = 284.3 meters (rounded to one decimal place)
Thus, the particle is approximately 284.3 meters from the origin after 8 seconds.