# Ice skater gliding

1. Jan 13, 2014

### negation

1. The problem statement, all variables and given/known data

A skater is gliding along the ice at 2.4 m/s, when she undergoes an acceleration of magnitude 1.1m/s^2 for 3.0s. At the end of that time she is moving at 5.7 m/s.

What must be the angle between the acceleration vector and the initial velocity vector?

3. The attempt at a solution

a = dv/dt

The issue I'm facing in circular motion is drawing a geometrical model. I don't know how to set it up geometrically. I can't progress further without being able to build a model and draw conclusions from there.

Last edited: Jan 13, 2014
2. Jan 13, 2014

### collinsmark

The skater is not moving in circular motion. I'm pretty sure you should assume that the acceleration vector has a constant direction.

The acceleration vector can have two components though: one component is parallel to the initial velocity vector and the other component is perpendicular to the initial velocity vector.

3. Jan 13, 2014

### negation

Do you think I could have a leg-up with a geometrical model for me to make sense?

4. Jan 13, 2014

### negation

Could you expound in the component being parallel to the initial velocity? Also, if the motion is non-circular, how could the acceleration be normal to the initial velocity?

5. Jan 13, 2014

### Simon Bridge

There is a way to discover the geometry for yourself.

If the acceleration were at 0deg, what would the final speed have been?
If the acceleration were at 180deg, what would the final speed have been?
If the acceleration were at 90deg, what would the final speed have been?

What sort of geometry did you use to figure the answers?

6. Jan 13, 2014

### negation

I haven't figure out any answer.

Let's see if I can proceed this procedurally.

On the assumption the motion is circular about the origin and counter-clockwise(Cartesian):

a(0π), velocity is in the +j direction.
a(π), velocity is in the +j direction
a(π/2), velocity is in the +i direction

7. Jan 13, 2014

### Simon Bridge

OK - you think best in terms of axis:

put the initial velocity along the y axis.

Then $\vec v(t=0) = v_0\hat\jmath$ where v0=2.4m/s

If the acceleration were at 0deg, then $\vec a = a\hat\jmath$

Where a=1.1m/s/s

- find the final speed after 3 seconds: $|\vec v(t=3)|$.

is that bigger than, smaller than, or equal to the 5.7m/s given in the problem?

8. Jan 13, 2014

### Staff: Mentor

The motion is not circular. Consider very strongly using cartesian coordinates. The components of the acceleration vector in cartesian coordinates are going to be constant, independent of time. Consider aligning the initial velocity vector with the unit vector in the x-direction, i. The acceleration vector is going to have constant components in the x and y directions. Your job is to determine the these components, subject to the constraint that the overall acceleration vector has a magnitude of 1.1m/sec^2.

9. Jan 13, 2014

### negation

does this make sense?

10. Jan 13, 2014

### negation

I'm using Cartesian but I'm still unsure as to how the geometrical model should be built

11. Jan 13, 2014

### negation

I'm having a really big problem with this. My conceptual understanding is pretty shaky given I'm doing a self-study before the semester opens.

12. Jan 13, 2014

### haruspex

Take the initial motion as being in the x direction.
Create unknowns to represent the acceleration components in the x and y directions.
You know the magnitude of the acceleration. What equation does that give you?
You know the duration of the acceleration. What will be the velocity changes in the x and y directions?
What are the new velocities in the x and y directions?
You know the final speed. What equation does that give you?

13. Jan 13, 2014

### Staff: Mentor

Are you saying you don't know what equations to use? What is the relationship between velocity and acceleration (in vector form)? What is the relationship between velocity and acceleration in cartesian component form?

Chet

14. Jan 13, 2014

### negation

x^2 + y^2 = |a|?

I'm really clueless.

15. Jan 13, 2014

### negation

Acceleration is the derivative of velocity.
I'm saying how should I intepret the question geometrically. My books is pretty much worthless at only 2 pages for circular motion.

16. Jan 14, 2014

### haruspex

To avoid confusion, let's label the x and y components of acceleration ax, ay. Your equation is wrong. On the left hand side you have squares of accelerations; on the right hand side you have an acceleration that's not squared. Equations should always have the same sort of thing on each side.
Please try to correct the equation and attempt my next question:

17. Jan 14, 2014

### Staff: Mentor

I'm not sure I understand what you mean by interpreting the question geometrically. But, OK, here goes. Initially the skater is moving in a straight line in the x directions. The skater's direction starts changing as soon as she starts her consstant acceleration, oriented at an angle of θ to the x direction. After accelerating for a long time, her velocity and trajectory will eventually be a straight line at the angle θ to the x axis.

You already stated that the derivative of the velocity is equal to the acceleration. All you need to do now is to translate this statement into the language of mathematics with an equation. First write it down in terms of the vectors. Then write it down as two equations, in terms of the x and y components of the vectors. If the acceleration is 1.1 m/sec^2 and it is oriented at an angle θ to the x axis, you can resolve the acceleration into its components in the x and y directions. In terms of θ, what are these components?

Chet

Last edited: Jan 14, 2014
18. Jan 14, 2014

### negation

What is the mathematical reasoning for having squares of acceleration on the left hand and non-squared acceleration on the right?

19. Jan 14, 2014

### negation

X-component: 1.1m/s^2 cos theta
Y-component: 1.1m/s^2 sin theta

20. Jan 14, 2014

### Staff: Mentor

Excellent. Now write down the equations I asked for in component form, in terms of θ. What we are looking for is:
dvx/dt=???
dvy/dt=?????

Chet