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Gala2k [10]
3 years ago
11

A particle moves along the x-axis. The velocity of the particle at time t is −2t+8 . What is the total distance traveled by the

particle from t=1 to t=6?
Mathematics
1 answer:
Tema [17]3 years ago
5 0

Answer:

13

Step-by-step explanation:

The velocity is given by:

\displaystyle v(t)= -2t + 8

And the total distance given the velocity is given by:

\displaystyle \int_{a}^{b}|v(t)|\, dt

So, the total distance traveled by the particle from t = 1 to t = 6 will be:

\displaystyle d=\int_{1}^{6}|-2t+8|\, dt

The absolute value tells us that:

\displaystyle |-2t+8| = \left\{        \begin{array}{ll}            -2t+8 & \quad x \leq 4 \\            -(-2t+8) & \quad x > 4\\         \end{array}    \right

So, split the integral into two parts:

\displaystyle d=\int_{1}^{4} (-2t+8)\, dt+\int_{4}^{6}(-(-2t+8))\, dt

Integrate:

\displaystyle d=(-t^2+8t)\Big|_{1}^{4}+(t^2-8t)\Big|_{4}^{6}

Evaluate:

\begin{aligned} d&=[(-4^2+8(4))-(-1^2+8)]+[(6^2-8(6))-(4^2-8(4)]\\ &= [16-7]+[-12-(-16)]\\ &=9+4 \\ &=13 \end{aligned}

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