Question

A particle moves along the x-axis with velocity given by v(t) = 10 sin(0.4t^2)/(t^2 - t + 3) for time 0 ≤ t ≤ 3.5. The particle is a position x = -5 at time t = 0.

(a) Find the acceleration of the particle at time t = 3.

(b) Find the position of the particle at time t = 3.

(c) Evaluate
`\int_(0)^(3.5) v(t) \;dt` and
`\int_(0)^(3.5) |v(t)| \;dt`. nterpret the meaning of each integral in the context of the problem.

(d) A second particle moves along the x-axis with position given by x₂(t) = t^2 - t for 0 ≤ t ≤ 3.5. At what time t are the two particles moving with the same velocity?

Answer 1

(a)

The acceleration is the derivative of velocity. Use the calculator to evaluate this numerical derivative..


`v'(t) \approx -2.118`

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(b)

x(t) is the position function, which is the antiderivative of velocity.
Via the fundamental theorem of calculus


`\int_0^3 v(t)\,dt = x(3) - x(0) \implies x(3) = x(0) + \int_0^3 v(t)\,dt \implies \\ \\ x(3) = -5 + \int_0^3 v(t)\,dt = -1.760`

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(c)


`\int_0^(3.5) v(t)\, dt \approx 2.844`

The above quantity represents the particle's displacement, change in position,
for 0 ≤ t ≤ 3.5.


`\int_0^(3.5) |v(t)|\, dt \approx 3.737`

The above quantity represents the particle's distance traveled on 0 ≤ t ≤ 3.5.

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(c)


`x_2(t) = t^2 - t \implies x_2'(t) = 2t - 1`


`x_2'(t) = v(t) \implies t \approx 1.571`

t ≈ 1.571