2. A particle moving in the xy-plane has position (𝑥(𝑡),𝑦(𝑡)) at time 𝑡≥0, where 𝑑𝑥𝑑𝑡=cos(𝑡2) and 𝑑𝑦𝑑𝑡=𝑒𝑡sin(𝑡2). At time 𝑡=0, the particle is at position (1,2). The figure above shows the path of the particle for 0≤𝑡≤2.(a) Find the position of the particle at time 𝑡=2.
Question
- A particle moving in the xy-plane has position (𝑥(𝑡),𝑦(𝑡)) at time 𝑡≥0, where 𝑑𝑥𝑑𝑡=cos(𝑡2) and 𝑑𝑦𝑑𝑡=𝑒𝑡sin(𝑡2). At time 𝑡=0, the particle is at position (1,2). The figure above shows the path of the particle for 0≤𝑡≤2.(a) Find the position of the particle at time 𝑡=2.
Solution
To find the position of the particle at time t=2, we need to integrate the given velocity functions and use the initial position as the constant of integration.
The x-coordinate of the position of the particle at time t is given by the integral of dx/dt from 0 to t. So, we have:
x(t) = ∫ cos(t^2) dt from 0 to t.
Unfortunately, this integral cannot be expressed in terms of elementary functions. However, we can use numerical methods to approximate it.
Similarly, the y-coordinate of the position of the particle at time t is given by the integral of dy/dt from 0 to t. So, we have:
y(t) = ∫ e^t sin(t^2) dt from 0 to t.
Again, this integral cannot be expressed in terms of elementary functions, but can be approximated using numerical methods.
So, to find the position of the particle at time t=2, we would need to approximate these two integrals.
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