Ice skater gliding

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Homework Statement



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?



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.
 
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  • #2
collinsmark
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Homework Statement



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?



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.
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.
 
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  • #3
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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.
Do you think I could have a leg-up with a geometrical model for me to make sense?
 
  • #4
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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.
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
Simon Bridge
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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
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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?
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
Simon Bridge
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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
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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
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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?
Capture.JPG


does this make sense?
 
  • #10
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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.
I'm using Cartesian but I'm still unsure as to how the geometrical model should be built
 
  • #11
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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
haruspex
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I'm using Cartesian but I'm still unsure as to how the geometrical model should be built
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
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I'm using Cartesian but I'm still unsure as to how the geometrical model should be built
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
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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?

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

I'm really clueless.
 
  • #15
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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


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

I'm really clueless.
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:
You know the duration of the acceleration. What will be the velocity changes in the x and y directions?
 
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  • #17
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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.
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
 
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  • #18
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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:
What is the mathematical reasoning for having squares of acceleration on the left hand and non-squared acceleration on the right?
 
  • #19
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I'm not sure I understand what you mean by interpreting the question numerically. 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
X-component: 1.1m/s^2 cos theta
Y-component: 1.1m/s^2 sin theta
 
  • #20
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X-component: 1.1m/s^2 cos theta
Y-component: 1.1m/s^2 sin theta
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
 
  • #21
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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
vx = [1.1m/s^2 cosΘ]3s
vy = [1.1m/s^2 sinΘ]3s

dvx/dt = 3.3ms^-1 d/dt cos Θ .dΘ/dt = -3.3ms^-1 sinΘ .ω
dvy/dt = 3.3ms^-1 d/dt cos Θ .dΘ/dt = 3.3ms^-1 cosΘ .ω
 
  • #22
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vx = [1.1m/s^2 cosΘ]3s
vy = [1.1m/s^2 sinΘ]3s

dvx/dt = 3.3ms^-1 d/dt cos Θ .dΘ/dt = -3.3ms^-1 sinΘ .ω
dvy/dt = 3.3ms^-1 d/dt cos Θ .dΘ/dt = 3.3ms^-1 cosΘ .ω
These equations are not correct, but it won't take much to correct them.

The kinematic equations I was asking for can be written as:
[tex]\frac{dv_x}{dt}=(1.1) \cosθ[/tex]
[tex]\frac{dv_y}{dt}=(1.1) \sinθ[/tex]

The right hand sides of these equations are constant. They are the components of the acceleration. The angle θ is constant, and doesn't depend on time. Does this make sense so far?

From your problem statement, what are the initial values of vx and vy at time t = 0? Do you know how to integrate the above differential equations from time t = 0 to arbitrary time t, subject to these initial conditions? If yes, please do so and show us your results for vx(t) and vy(t).

Chet
 
  • #23
haruspex
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What is the mathematical reasoning for having squares of acceleration on the left hand and non-squared acceleration on the right?
You wrote
x^2 + y^2 = |a|
Since that was in response to my question about x and y components of acceleration, I presume that's what the x and y refer to there. On the left hand side you have x^2 and y^2, so those are each the square of an acceleration. In dimensional terms, L2T-4 (length squared over time to the fourth). On the right hand side you have just an acceleration, LT-2. That cannot be right.
This is a really useful test to apply to your equations. If they are inconsistent dimensionally then they must be wrong.
 
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  • #24
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These equations are not correct, but it won't take much to correct them.

The kinematic equations I was asking for can be written as:
[tex]\frac{dv_x}{dt}=(1.1) \cosθ[/tex]
[tex]\frac{dv_y}{dt}=(1.1) \sinθ[/tex]

The right hand sides of these equations are constant. They are the components of the acceleration. The angle θ is constant, and doesn't depend on time. Does this make sense so far?

From your problem statement, what are the initial values of vx and vy at time t = 0? Do you know how to integrate the above differential equations from time t = 0 to arbitrary time t, subject to these initial conditions? If yes, please do so and show us your results for vx(t) and vy(t).

Chet

Yes, your equation make sense. However, why doesn't mine? in taking ax and multiplying by time, I obtain velocity.

The initial velocity is (2.4ms^-1, 0ms^-1)

Yes I do know how to integrate dvx/dt and dvy/dt
 
  • #25
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Yes, your equation make sense. However, why doesn't mine? in taking ax and multiplying by time, I obtain velocity.
In your equation for vx, you left out the initial velocity of the skater. Otherwise, your first two equations are correct. Your second two equations (taking derivatives of the velocities) were not done correctly, and were not even necessary for solving this problem. Once you make the correction to the equation for vx, take the sum of the squares of the x and y components of velocity and set them equal to the magnitude of the final velocity you are seeking. This will give you an equation for solving for the angle θ (or sinθ or cosθ).

Chet
 
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