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: $x (t) = \sin (2 t) + t$

Answer the following:

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

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

See development and solution

Development:

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

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

$Δ x = 3600 m$

We find the average speed in this interval: $v_{m} = \frac{Δ x}{Δ t} = \frac{3600 m}{3600 s} = 1 m / s$

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

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

Therefore,

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

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

Solution:

a) $v_{m} = 1 m / s$

$Δ x = 3600 m$

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

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

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