A particle at point A is 50 mm away from a second particle at point B. The first particle is moving toward point B at a constant rate and the second particle is moving at a right angle to the line AB at a rate that is 1/3 of the rate of the first particle. The particles get closer to each other for 3/4 of a second, and then begin to get further apart. How fast is the second particle moving in mm/sec?
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To find the speed of the second particle, we need to first understand the motion of both particles and then apply the appropriate formulas.
Let's denote the speed of the first particle as v₁ and the speed of the second particle as v₂.
From the given information, we know that the first particle is initially 50 mm away from the second particle. Since it is moving towards the second particle at a constant rate, we can say its distance to the second particle is decreasing at a constant rate of v₁ mm/sec.
The second particle is moving at a right angle to the line AB. This means that its motion does not affect the distance between the particles, but rather the separation between them is determined by the motion of the first particle. Therefore, we can ignore the second particle's motion and focus on the first particle.
During the time interval when the particles are getting closer (3/4 of a second), the distance between them decreases. We can express this by using the formula:
Distance = Speed × Time
Since the speed of the first particle is v₁ mm/sec and the time interval is 3/4 of a second, the distance between the particles decreases by:
d₁ = v₁ × (3/4)
Now, we know that the particles begin to get further apart after this interval. This means the distance between them increases, but only as a result of the first particle's motion.
Therefore, we can express the change in distance by using the formula:
Change in Distance = Speed × Time
Since the speed of the first particle is v₁ mm/sec and the time interval is 1 second, the distance between the particles increases by:
d₂ = v₁ × 1
We know that the change in distance during the interval when the particles are getting further apart is equal to the initial distance minus the distance after the particles get closer:
d₂ - d₁ = 50 (initial distance)
Substituting the expressions for d₁ and d₂, we can solve for v₁:
v₁ × 1 - v₁ × (3/4) = 50
Simplifying the equation gives:
v₁/4 = 50
Multiply both sides by 4 to isolate v₁:
v₁ = 200 mm/sec
Now, to find the speed of the second particle, v₂, we are given that it moves at a rate that is 1/3 of the rate of the first particle. Therefore:
v₂ = (1/3) × v₁
Substituting the value of v₁, we find:
v₂ = (1/3) × 200
Simplifying, we get:
v₂ = 66.67 mm/sec
Therefore, the second particle is moving at a speed of approximately 66.67 mm/sec.