Question

A model rocket fired from the ground
ascends with constant upward acceleration.
After 1.0 s from firing a small bolt is
dropped from the rocket and after 5.0 s from
firing, its fuel is then finished. The bolt
strikes the ground after 2.0 s from the instant
it was dropped. Acceleration due to gravity
is g = 10 m/s2
.
(a) acceleration of the rocket while running
on its fuel is 8.0 m/s2
.
(b) rocket was at height 100 m above the
ground when its fuel was finished.
(c) maximum speed of the rocket during its
flight is 40 m/s
(d) total airtime of the rocket is 15 s
THE QUESTION IS HOW MANY STATEMENTS ARE TRUE?
CAN U PLEASE EXPLAIN IN DETAIL

Answer 1

Answer on Question #60022, Physics – Mechanics | Relativity

A model rocket fired from the ground ascends with constant upward acceleration.

After 1.0 s from firing a small bolt is dropped from the rocket and after 5.0 s from firing, its fuel is then finished. The bolt strikes the ground after 2.0 s from the instant it was dropped. Acceleration due to gravity is $g = 10 \, \text{m/s}^2$.

(a) acceleration of the rocket while running on its fuel is $8.0 \, \text{m/s}^2$

(b) rocket was at height 100 m above the ground when its fuel was finished.

(c) maximum speed of the rocket during its flight is $40 \, \text{m/s}$

(d) total airtime of the rocket is 15 s

THE QUESTION IS HOW MANY STATEMENTS ARE TRUE?

CAN U PLEASE EXPLAIN IN DETAIL

Solution:

Equations for rocket:

$$\begin{array}{l} h_1 = \frac{a t_1^2}{2} \\ a = \frac{v_1}{t_1} \end{array}$$

where initial velocity $v_0 = 0$, $t_1 = 1.0$ s and $a$ is acceleration.

Substituting we have

$$h_1 = \frac{v_1 t_1}{2}$$

Equations for bolt:

$$h = h_1 + v_1 t_2 - \frac{g t_2^2}{2}$$

where initial velocity $v_1$, $t_2 = 2.0$ s and final $h = 0$.

We obtain system of equations:

$$\left\{ \begin{array}{c} h_1 = \frac{v_1}{2} \\ h_1 + 2 v_1 - 20 = 0 \end{array} \right.$$ $$\begin{array}{l} \frac{v_1}{2} = 20 - 2 v_1 \\ v_1 = 8 \frac{\text{m}}{\text{s}} \end{array}$$

Thus,

$$a = \frac{v_1}{t_1} = \frac{8}{1} = 8 \, \text{m/s}^2$$

(a) acceleration of the rocket while running on its fuel is $8.0 \, \text{m/s}^2$ is TRUE

$$h_5 = \frac{a t_5^2}{2} = 8 \cdot \frac{5^2}{2} = 100 \, \text{m}$$

(b) rocket was at height $100 \, \text{m}$ above the ground when its fuel was finished is TRUE.

The maximum speed

$$v = a t _ {5} = 8 \cdot 5 = 40 \mathrm{~m/s}$$

(c) maximum speed of the rocket during its flight is $40\mathrm{m/s}$ is TRUE.

The airtime of the rocket after the fuel was finished

$$h = h _ {5} + v _ {5} t - \frac {g t ^ {2}}{2}$$ $$h _ {5} = 100 \mathrm{~m}$$ $$v _ {5} = 40 \frac {\mathrm{m}}{\mathrm{s}}$$ $$h = 0$$ $$5 t ^ {2} - 40 t - 100 = 0$$ $$t = 10 \mathrm{~s}$$

Thus, the total time is

$$\text{total time} = t _ {5} + t = 5 + 10 = 15 \mathrm{~s}$$

(d) total airtime of the rocket is 15 s is TRUE.

Output: a, b, c, d are TRUE.

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