Question

An object thrown in the air has a velocity after seconds that can be described by v(t) = - 9.8z + 24(m) meters/second) and a height h(t) = - 4.9t ^ 2 + 2t + 60 in meters). The object has n = 2 kilograms. The kinetic energy of the shirst is given by K= 1 2 mr^ 2 , and the potential energy is given by 19.5mh. Find an expression for the total kinetic and potential energy K+ U as a functim of time. What does this expression tell you about the energy of the falling object?

Answer 1

The total energy is an illustration of a composite function.

• The function of total energy is $(K + P)(t) = 2.94t^2 -432.4t + 1716$.
• The expression tells the total energy of the falling object.

Given

$v(t) = -9.8t + 24$

$h(t) = -4.9t^2 + 2t + 60$

$m =2$ -- the mass of the object

First, we calculate the function of kinetic energy

$K = \frac 12mv^2$

So, we have:

$K(t) = \frac 12m \times (-9.8t + 24)^2$

Substitute $m =2$

$K(t) = \frac 12 \times 2 \times (-9.8t + 24)^2$

$K(t) = (-9.8t + 24)^2$

$K(t) = 96.04t^2 - 470.4t + 576$

The function of potential energy is then calculated as:

$P = 9.5mh$, not 19.5mh

So, we have:

$P(t) = 9.5m \times (-4.9t^2 + 2t + 60)$

Substitute $m =2$

$P(t) = 9.5 \times 2 \times (-4.9t^2 + 2t + 60)$

$P(t) = 19 \times (-4.9t^2 + 2t + 60)$

$P(t) = -93.1t^2 + 38t + 1140$

So, the composite function for the total energy is:

$(K + P)(t) = K(t) + P(t)$

$(K + P)(t) = 96.04t^2 - 470.4t + 576 -93.1t^2 + 38t + 1140$

Collect like terms

$(K + P)(t) = 96.04t^2 -93.1t^2 - 470.4t+ 38t + 576 + 1140$

$(K + P)(t) = 2.94t^2 -432.4t + 1716$

The expression tells the total energy of the falling object.

Read more about composite functions at:

https://brainly.com/question/8308119