Three particles move along the x-axis, each starti...

Question

Three particles move along the x-axis, each starting at t₀ = 0 s. The graph for A is a position-versus-time graph; the graph for B is a velocity-versus-time graph; the graph for C is an acceleration-versus-time graph.

(a) Find the velocity of the particle A at exttip{t}{t} = 5.5 s. The graph shows x-position of the particle A as a function of time. Time is measured from 0 to 8 seconds on the x-axis. Position is measured from -30 to 30 meters on the y-axis. Position keeps the constant value of 30 meters over the course of the first 2 seconds. It drops linearly from 30 to -30 meters at the time interval of 2 to 8 seconds.

(b) Find the velocity of the particle B at exttip{t}{t} = 5.5 s. The graph shows x-component of the velocity for the particle B as a function of time. Time is measured from 0 to 8 seconds on the x-axis. Velocity is measured from -30 to 30 meters per second on the y-axis. Velocity keeps the constant value of 30 meters per second over the course of the first 2 seconds. It drops linearly from 30 to -30 meters per second at the time interval of 2 to 8 seconds.

(c) Find the velocity of the particle C at exttip{t}{t} = 5.5 s. Particle starts with v_0x = 10 m/s at t₀ = 0 s.

Patrick Ford · Accepted Answer

In this problem, we have position-time, velocity-time, and acceleration-time graphs. Let's check out how to determine the velocity at a given time from each graph.(a)In a position-time graph, velocity is given by the slope of the curve.[readmore]At t = 5.5s, the slope is a straight line. Determining the slope of a straight line is simple and straightforward. slope = Δx/ΔtWe'll take the slope from points (5s,0m) to point (6s,-10m).v = slope = Δx/Δt = (-10 - 0)/(6 - 5) = -10m/sThe velocity of particle A at 5.5s is -10m/s(b) In a velocity-time graph, we don't need to perform any calculations to determine the velocity.From the graph, the velocity of particle B at 5.5s is -5m/s(c)In acceleration time graphs, the change in velocity is the area under the graph. From 0-2s, the area is a rectangle. From 2-5s, the area is a triangle. From 5-5.5s, the area is a triangle.Areas above the t-axis are positive because the height ax is positive. Areas below the t-axis are negative because the height ax is negative.Δv = A1 + A2 + A3 = (ax)(Δt) + (1/2)(Δax)(Δt) + (1/2)(ax)(Δt) = (30)(2 - 0) + (1/2)(30)(5 - 2) - (1/2)(-5)(5.5 - 5) = 60 + 45 - 1.25 = 104 m/sv = v0 + at = v0 + Δv = 10 + 104 = 114 m/sThe velocity of particle C at 5.5s is 114m/s

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