Question

Please help me understand I am confused

Answer 1

9514 1404 393

Explanation:

a) The velocity curve is linearly increasing from 0 to 6 m/s over a period of 2 seconds, then linearly decreasing from 6 m/s to 0 over the same period. The acceleration is the rate of change of velocity, so for the first half of the motion the acceleration is a constant (6 m/s)/(2 s) = 3 m/s². Similarly, over the second half of the motion, the acceleration is a constant (-6 m/s)/(2 s) = -3 m/s².

The distance traveled is the integral of the velocity, so the linearly increasing velocity will cause the distance vs. time curve to have a parabolic shape. The shape will likewise be parabolic, but with decreasing slope, as the velocity ramps down to zero. Overall, the distance versus time curve will have an "S" shape.

The motion (position and velocity) will be continuous, but the acceleration will not be. There will be a significant "j.erk" at the 2-second mark where acceleration abruptly changes from increasing the velocity to braking (decreasing the velocity).

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b) The attachment shows the (given) velocity curve in meters per second and its integral, the position curve, in meters.

The integral in the attached works nicely for machine evaluation. For hand evaluation, it is perhaps best written piecewise:

  $s(t)=\begin{cases}\displaystyle\int_0^t{3x}\,dx\qquad\text{for x\le2 }\\\\\displaystyle6+\int_2^t{(12-3x)}\,dx\qquad\text{for 2$