Which steps did I do wrong in this question?

Question

Question: Find V₁

mgh₁+(1/2)mv₁² = (1/2)mv₂²

m = 22000

v₂ = 4871

h₁ = 505000

g = 9.81

My Solution:

1. mgh₁+(1/2)mv₁²=(1/2)mv₂²

2. (22000)(9.81)(505000) + (1/2)(22000)v₁² = (1/2)(22000) (4871²)

3. 108989100000 + 11000v₁² = 260993051000

4. 11000v₁² = 260993051000 - 108989100000

5. 11000v₁² = 152003951000

6. v₁² = 152003951000/11000

7. v₁² = 13818541

8. v₁ = 3717.33

When Used:

1. (22000)(9.81)(505000) + (1/2)(22000)(3717.33²) = (1/2)(22000) (4871²)

2. 108989100000 + 152003965617.9 = 260993051000

3. 260993065617.9 = 260993051000

4. 260993065617.9 Doesn't Equal 260993051000

Answer 1

Let $m = 22000$ and $v_2 = 4871$, $h_1 = 505000$, $g = 9.81$ be the constants known in physics. You have then the equation

$$mgh_1 + \frac{mv_1^2}{2} = \frac{mv_2^2}{2}$$

relating the mechanical energy in the instants $1$ and $2$, because of energy conservation. Then you'll have

$$v_1^2 = v_2^2-2gh_1 \implies v_1 = \sqrt{v_1 ^2-2gh_1}$$

If you just set the numbers on it

$$v_1 = \sqrt{13818541} \approx 3717.33$$

So in my opinion you didn't make incorrect steps but instead just got lost in your notation approach.

Answer 2

Well, in general we have:

$$\text{m}\cdot\text{g}\cdot\text{h}_1+\frac{\text{m}\cdot\text{v}_1^2}{2}=\frac{\text{m}\cdot\text{v}_2^2}{2}\tag1$$

Now, factor out $\text{m}$:

$$\text{m}\cdot\left\{\text{g}\cdot\text{h}_1+\frac{\text{v}_1^2}{2}\right\}=\text{m}\cdot\frac{\text{v}_2^2}{2}\tag2$$

Divide both sides by $\text{m}$:

$$\text{g}\cdot\text{h}_1+\frac{\text{v}_1^2}{2}=\frac{\text{v}_2^2}{2}\tag3$$

Put $\text{g}\cdot\text{h}_1$ to the other side:

$$\frac{\text{v}_1^2}{2}=\frac{\text{v}_2^2}{2}-\text{g}\cdot\text{h}_1\tag4$$

Multiply both sides by $2$:

$$2\cdot\frac{\text{v}_1^2}{2}=\text{v}_1^2=2\cdot\left\{\frac{\text{v}_2^2}{2}-\text{g}\cdot\text{h}_1\right\}\tag5$$

Take the square root of both sides:

$$\text{v}_1=\pm\space\sqrt{2\cdot\left\{\frac{\text{v}_2^2}{2}-\text{g}\cdot\text{h}_1\right\}}=\pm\space\sqrt{2}\cdot\sqrt{\frac{\text{v}_2^2}{2}-\text{g}\cdot\text{h}_1}\tag6$$

---

So, we get:

$$\text{v}_1=\pm\space\sqrt{2}\cdot\sqrt{\frac{4871^2}{2}-\frac{981}{100}\cdot505000}=\pm\space\sqrt{13818541}\approx\pm\space3717.329821\tag6$$