Question

the accompanying figure shows the velocity v = \\(\frac{ds}{dt}=f(t)\\) (m/sec) of a body moving along a coordinate line.
a. when does the body reverse direction?
b. when is it moving at a constant speed?
c. graph the bodys speed for 0 ≤ t ≤ 10.
d. graph the acceleration, where defined.
the body reverses direction at t = 2,4 (use a comma to separate answers as needed.)
the body moves at a constant speed at 5 ≤ t ≤ 8
choose the correct graph of the bodys speed for 0 ≤ t ≤ 10

Explanation:

Step1: Identify direction - change points

The body reverses direction when velocity changes sign. From the graph, velocity changes sign at \(t = 2\) and \(t=4\).

Step2: Identify constant - speed intervals

Speed is the absolute - value of velocity. The body has a constant speed when the magnitude of velocity is constant. From the graph, \(|v|\) is constant for \(5\leq t\leq8\).

Step3: Graph the speed

Speed \(s(t)=|v(t)|\). For \(0\leq t\leq2\), \(v(t)\geq0\), so \(s(t)=v(t)\); for \(2 < t<4\), \(v(t)<0\), so \(s(t)= - v(t)\); for \(4 < t\leq5\), \(v(t)\geq0\), so \(s(t)=v(t)\); for \(5\leq t\leq8\), \(v(t)\) is constant, so \(s(t)\) is constant; for \(8 < t\leq10\), \(v(t)\) is decreasing, so \(s(t)=|v(t)|\). The correct graph of speed is the one where the values are non - negative (since speed is non - negative) and match the magnitude of the velocity graph. The correct graph is C as speed is always non - negative and has constant values when velocity is constant in magnitude.

Step4: Graph the acceleration

Acceleration \(a(t)=\frac{dv}{dt}\). The acceleration is the slope of the velocity - time graph. For \(0 < t<2\), \(a(t)>0\); at \(t = 2\), \(a(t)\) has a non - continuous change (a sharp turn in the velocity graph); for \(2 < t<4\), \(a(t)<0\); at \(t = 4\), \(a(t)\) has a non - continuous change; for \(4 < t<5\), \(a(t)>0\); for \(5 < t<8\), \(a(t)=0\) (since \(v\) is constant); for \(8 < t<10\), \(a(t)<0\).

Answer:

a. \(t = 2,4\)
b. \(5\leq t\leq8\)
c. C
d. (Graph of acceleration: positive slope for \(0 < t<2\), negative slope for \(2 < t<4\), positive slope for \(4 < t<5\), zero slope for \(5 < t<8\), negative slope for \(8 < t<10\) with non - continuous changes at \(t = 2\) and \(t = 4\))