Solving Vectors & Surfing Problem at Waikiki

[Eugen]

Hello,

Here's the problem:

Suppose we orient the x-axis of a two-dimensional coordinate system along the beach at Waikiki. Waves approaching the beach have a velocity relative to thTe shore given by v = (1.3 m/s)y. Surfers move more rapidly than the waves, but at an angle to the beach. The angle is chosen so that the surfers approach the shore with the same speed as the waves. (a) If a surfer has a speed of 7.2 m/s relative to the water, what is her direction of motion relative to the positive x axis? (b) What is the surfer’s velocity relative to the wave? (c) If the surfer’s speed is increased, will the angle in part (a) increase or decrease? Explain.

The problem says that "the surfers approach the shore with the same speed as the waves". I take this to mean "the surfers velocity relative to the shore is the same as the waves velocity relative to the shore: 1.4 m/s". I think that's impossible because the surfers velocity relative to the shore is the waves velocity relative to the shore (1.4 m/s) + the surfers velocity relative to the water (7.2 m/s). Not matter what the angle of the surfers motion is, their speed relative to the shore will be much larger than 1.4 m/s.
Am I missing something?

 

[berkeman]

Hello,

Here's the problem:

Suppose we orient the x-axis of a two-dimensional coordinate system along the beach at Waikiki. Waves approaching the beach have a velocity relative to thTe shore given by v = (1.3 m/s)y. Surfers move more rapidly than the waves, but at an angle to the beach. The angle is chosen so that the surfers approach the shore with the same speed as the waves. (a) If a surfer has a speed of 7.2 m/s relative to the water, what is her direction of motion relative to the positive x axis? (b) What is the surfer’s velocity relative to the wave? (c) If the surfer’s speed is increased, will the angle in part (a) increase or decrease? Explain.

The problem says that "the surfers approach the shore with the same speed as the waves". I take this to mean "the surfers velocity relative to the shore is the same as the waves velocity relative to the shore: 1.4 m/s". I think that's impossible because the surfers velocity relative to the shore is the waves velocity relative to the shore (1.4 m/s) + the surfers velocity relative to the water (7.2 m/s). Not matter what the angle of the surfers motion is, their speed relative to the shore will be much larger than 1.4 m/s.
Am I missing something?

Welcome to the PF.

(In the future, please fill out the Homework Help Template you are provided when posting a schoolwork question. It makes it easier to answer your questions)

The surfer is moving toward the shore at the same speed as the wave (in the y direction). The surfer is moving in the x direction parallel to the shore with a different speed, and the vector sum of the two is the 7.2m/s with respect to a fixed spot (attached to the Earth).

 

[berkeman]

BTW, in your problem statement the y-speed was stated as 1.3m/s, and then it looks like you changed it to 1.4m/s?

 

[Eugen]

Thanks for answering. :) It's 1.3 m/s all the time...

The surfer is moving toward the shore at the same speed as the wave (in the y direction). The surfer is moving in the x direction parallel to the shore with a different speed, and the vector sum of the two is the 7.2m/s with respect to a fixed spot (attached to the Earth).

This seems clear enough, but the problem says: "If a surfer has a speed of 7.2 m/s relative to the water,...". So, the 7.2 m/s speed is relative to the water, not to the shore. If the surfer moves parallel to the beach with 7.2 m/s (relative to the water) he can't move with 1.3 m/s relative to the beach. This is what I don't understand.

 

[berkeman]

Thanks for answering. :) It's 1.3 m/s all the time...
This seems clear enough, but the problem says: "If a surfer has a speed of 7.2 m/s relative to the water,...". So, the 7.2 m/s speed is relative to the water, not to the shore. If the surfer moves parallel to the beach with 7.2 m/s (relative to the water) he can't move with 1.3 m/s relative to the beach. This is what I don't understand.

But the water is not moving in the x-y plane. Only the waves and the surfer are moving in the x-y plane. Waves are a deflection in the z direction.

If there is a beach ball floating on the water (no wind) as the surfer goes by, does the beach ball move toward the shore? (Well, there is a small oscillation in the y-direction, but overall it does not move).

 

[Eugen]

If the water is not moving, then the surfer's speed relative to the water should be the same as his speed relative to the shore (vector sum of the waves speed and surfer's horizontal speed). Or that is not so?
Then how can it be that " the surfers approach the shore with the same speed as the waves."

 

[berkeman]

If the water is not moving, then the surfer's speed relative to the water should be the same as his speed relative to the shore (vector sum of the waves speed and surfer's horizontal speed). Or that is not so?

Yes, his V(x,y) is the same with respect to the still water and the shore.

Then how can it be that " the surfers approach the shore with the same speed as the waves."

His V(y) is the same as the waves' V(y) while he is riding the wave into shore.

His motion V(x,y) has two components. One component is 1.3m/s in the y direction, and his x component combines vectorially to make his overall velocity vector V(x,y) = 7.2m/s with respect to the still water and still Earth. Can you write the vector equation that results in the final 7.2m/s?

 

[Eugen]

Yes, that equation is V_s,sh = V_w,sh + V_s,w where w = waves, sh = shore, s = surfer.
V_s,sh = 7.2 m/s and V_w,sh = 1.3 m/s.
But I still don't understand how can the surfer approach the shore with the same speed as the waves. The surfer's speed relative to the shore is 7.2 m/s. The waves speed relative to the shore is 1.3 m/s.

 

[berkeman]

Yes, that equation is V_s,sh = V_w,sh + V_s,w where w = waves, sh = shore, s = surfer.
V_s,sh = 7.2 m/s and V_w,sh = 1.3 m/s.
But I still don't understand how can the surfer approach the shore with the same speed as the waves. The surfer's speed relative to the shore is 7.2 m/s. The waves speed relative to the shore is 1.3 m/s.

Just sketch the vectors and add them.

 

[Eugen]

Just sketch the vectors and add them.

I can't, because I don't know how to find out the x and y components of V_s,w and V_s,sh.

 

[berkeman]

I can't, because I don't know how to find out the x and y components of V_s,w and V_s,sh.

Draw an x-y coordinate system, with the origin at the shore (sand is positive y and water is negative y). Positive x is to the right and positive y is up.

Then draw the wave at y = -30m (or whatever). That's a horizontal line going from left to right. Draw a dot at x = 0 on that wave line, and that's your surfer.

Draw a vertical velocity vector in the y direction of 1.3m/s. That's the velocity of the wave and surfer in the y direction (towards the beach). Then draw a longer velocity vector in the +x direction along the wave line -- that is the component of the surfer's velocity parallel to the wave. Then draw a resultant vector from adding the x and y components of his velocity -- it will point up and to the right, and will have a magnitude of 7.2m/s. You should be able to use the Pythagorean theorem to figure out the x component of the surfer's velocity from this sketch...

 

[Eugen]

I didn't realize that the surfer's movement relative to waves is perpendicular on the waves movement. Is this related to the surfer approaching the shore with the same speed as the waves? If the surfer moved at some angle with respect to the waves, wouldn't he approach the shore at the same speed as the waves?
Put this way, the problem is very simple. The answer at point a) is 10 °.