# Problems from Physical interpretation of the derivative

A radar detects the position of a ship at every moment therefore one can know the trajectory of the ship, which turns out to be: $$x(t)=\sin (2t) +t$$

a) Find the average speed and the distance where the ship is after the first hour of trajectory ($$1h=3600s$$)

b) Find the instantaneous speed when $$t=10s$$ and $$t=100s$$

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### Development:

a) To compute the covered distance we are going to see the initial and final positions.

$$x(0)=0m; \ x(3600s)=3600m$$

$$\Delta x=3600m$$

We find the average speed in this interval: $$v_m=\dfrac{\Delta x}{\Delta t}=\dfrac{3600m}{3600s}=1m/s$$\$

b) We can compute the generic instantaneous speed and then we will substitute this for the required moments.

$$v(t)=x'(t)=2 \cos(2t)+1$$

Therefore,

$$v(10s)=2,88 \ m/s$$

$$v(100s)=-0,88 \ m/s$$

### Solution:

a) $$v_m=1m/s$$

$$\Delta x=3600m$$

b) $$v(10s)=2,88 \ m/s$$

$$v(100s)=-0,88 \ m/s$$

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