Solving for Speed: Exploring Velocity and Magnitude

[Svelte1]

Homework Statement

Okay, I saw a derivation of work equaling a change in kinetic energy and it explicitly states that this is regarding the MAGNITUDE of the velocity vector, however in an example question the formula is used for the horizontal velocity component and I don't understand how this can be done, seems incorrect to me.

Relevant Equations

Work= change in potential energy=change in kinetic energy

https://ibb.co/jG6n0jZ
The 15 is fine as this is clearly his overall magnitude but then v2 is equated to the horizontal velocity rather than the magnitude.

 

[kuruman]

At the bottom of the incline the motion is in the horizontal direction and there is no vertical velocity component. This means that the magnitude of the velocity (speed) is the same as the horizontal component.

On edit: In other words (and an equation), the magnitude of the velocity, also known as speed, is$$v=\sqrt{v_x^2+v_y^2}.$$ When $v_y=0$, $v=v_x$.

 

[Lnewqban]

At the bottom of the incline the motion is in the horizontal direction and there is no vertical velocity component. This means that the magnitude of the velocity (speed) is the same as the horizontal component.

I believe that the picture is misleading, as it incorrectly shows position #3 at a higher level, unless the skier is jumping just when arriving at point #2.

 

[kuruman]

I believe that the picture is misleading, as it incorrectly shows position #3 at a higher level, unless the skier is jumping just when arriving at point #2.

The statement of the problem includes "$\dots~$ they jump upward, achieving a vertical velocity component of 3 m/s." You must have missed it.

 

[Lnewqban]

The statement of the problem includes "$\dots~$ they jump upward, achieving a vertical velocity component of 3 m/s." You must have missed it.

I indeed missed it!
Thank you.

 

[Svelte1]

Thanks!

If we use the eq. before they jump then we have 24.8 for horizontal and magnitude of velocity, Then as they leave the ramp I can solve for 25 magnitude.

I still have a slight problem with the general model though. If we use the same equation for when they start to jump, but before they change their height, they have gained a vertical velocity, but we end with 24.8 for the magnitude.

 

[kuruman]

Thanks!

If we use the eq. before they jump then we have 24.8 for horizontal and magnitude of velocity, Then as they leave the ramp I can solve for 25 magnitude.

I still have a slight problem with the general model though. If we use the same equation for when they start to jump, but before they change their height, they have gained a vertical velocity, but we end with 24.8 for the magnitude.

When the jump has just started, the skier’s muscles have added a vertical component to his existing horizontal velocity by pushing against the surface. That increases the speed and hence the kinetic energy. When skied reaches maximum height, the vertical component is zero and the kinetic energy is back to what it was before the jump.

 

[Svelte1]

I suppose they can't gain the vertical velocity without a change in their height either, so I was wrong with what I said. Even if its just an almost infinitesimal amount of distance traveled upwards..

 

[kuruman]

I suppose they can't gain the vertical velocity without a change in their height either, so I was wrong with what I said. Even if its just an almost infinitesimal amount of distance traveled upwards..

You got it backwards. The height does not cause the vertical velocity. The skier cannot gain height unless a force adds to the existing kinetic energy in such a way as to add a vertical velocity component. That force is provided by the skier's muscles pushing against the surface. The cause is the addition of the vertical component of the velocity that increases the speed and hence the kinetic energy. The effect is the gain in height. When the skier reaches maximum height, the work done by gravity on the skier has taken away the energy that the skier's muscles added while executing the jump.

 

[Svelte1]

I just meant he can't have been said to have any vertical velocity until he has gained some vertical distance.

 

[kuruman]

I just meant he can't have been said to have any vertical velocity until he has gained some vertical distance.

That is true. The velocity is the rate of change of position with respect to time. If the skier is still in contact with the surface and you know that the velocity has acquired a vertical component at that moment, then you can predict that the skier will be off the surface at the next moment. If you don't know that the skier has acquired a vertical component at that given moment and you see that the skier is off the surface at the next moment, you can infer that the skier has acquired a vertical component at the earlier moment. Acquiring a vertical component is a necessary and sufficient condition for the skier to be off the surface.