Question

a particle moves along the $x$-axis. the function $v(t)$ gives the particles velocity at any time $t \geq 0$:
$v(t) = t^2 + 2t + 5$
what is the particles velocity $v(t)$ at $t = 5$
$v(5) = \square$
what is the particles acceleration $a(t)$ at $t = 5$?

Explanation:

Step1: Substitute t=5 into v(t)

$v(5) = 5^2 + 2(5) + 5$

Step2: Calculate v(5)

$v(5) = 25 + 10 + 5 = 40$

Step3: Find acceleration a(t) (derivative of v(t))

$a(t) = v'(t) = 2t + 2$

Step4: Substitute t=5 into a(t)

$a(5) = 2(5) + 2$

Step5: Calculate a(5)

$a(5) = 10 + 2 = 12$

Answer:

$v(5) = 40$
$a(5) = 12$