Question

If the swimmer starts at rest, slides without friction, and descends through a vertical height of 2.41 m

, what is her speed at the bottom of the slide?

Answer 1

6.88 m/s

Step-by-step explanation:

The Conservation of Energy states that:

Initial Kinetic Energy + Initial Potential Energy = Final Kinetic Energy + Final Potential Energy

So we can write

`mgh_(i)+(1)/(2)mv_(i) ^(2)=mgh_(f)+(1)/(2)mv_(f) ^(2)`

We can cancel the common factor of
`m` which leaves us with

`gh_(i)+(1)/(2)v_(i) ^(2)=gh_(f)+(1)/(2)v_(f) ^(2)`

Lets solve for
`v_f`

`gh_(i)+(v_(i) ^(2))/(2)=gh_(f)+(v_(f) ^(2))/(2)`

Subtract
`gh_f` from both sides of the equation.

`gh_(i)+(v_(i) ^(2))/(2)-gh_(f)=(v_(f) ^(2))/(2)`

Multiply both sides of the equation by 2.

`2(gh_(i)+(v_(i) ^(2))/(2)-gh_(f))={v_(f) ^(2)`

Simplify the left side.

Apply the distributive property.

`2(gh_(i))+2(v_(i) ^(2))/(2)+2(-gh_(f))={v_(f) ^(2)`

Cancel the common factor of 2.

`2gh_(i)+v_(i) ^(2)-2gh_(f)={v_(f) ^(2)`

Take the square root of both sides of the equation to eliminate the exponent on the right side.

`{v_(f)=\sqrt{2gh_(i)+v_(i) ^(2)-2gh_(f)}`

We are given
`g,v_(i),h_(i),h_(f)`.

We can now solve for the final velocity.

`{v_(f)=\sqrt{(2*9.81*2.41)+(0^(2))-(2*9.81*0)`

Anything multiplied by 0 is 0.

`{v_(f)=\sqrt{2*9.81*2.41`

`{v_(f)=\sqrt{47.2842`

`v_f=6.88`