What is the Velocity of the Ball Leaving a Racket Moving at 60 mph?

[mathguy2]

Homework Statement

A player swinging a tennis racket comprises a very heavy flat object (the racket-arm body combination) which makes a head-on collision with a very light object (the 2 ounce ball). The ball leaves the racket according to the formula:

V_ball = ε × V_racket

The United States Tennis Association imposes a rule on ball manufacturers:

σ = 1.5

If the racket is moving at 60 mph, how fast does the ball come off the racket? How fast would a player have to swing the racket in order for the serve to come off the racket at 120 mph? A superball will come off a heavy racket with

ε = 1.9

How fast does it come off a racket moving at 70 mph?

Homework Equations

All are stated.

The Attempt at a Solution

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I'm having trouble conceptualizing this problem. In addition, the added information, σ = 1.5 confuses me because that symbol, σ, appears nowhere in the equation. Also, how many solutions are required? 3? The way I see it, I need to see how fast the ball comes off the racket if the racket is moving at 60mph (solution 1). I have to see how fast a player would have to swing the racket in order for the serve to come off the racket at 120 mph (solution 2). And I have to figure out how fast the ball comes off a racket moving at 70 mph (solution 3).

The added information about the "superball" is really confusing me as well. Is the weight of the ball 1.9 or is the weight of the ball 2.0? I'm very confused about the constants I should be using.

So for instance, for the first part of the question: "If the racket is moving at 60 mph, how fast does the ball come off the racket?"

V_ball = ε × V_racket

V_ball = ε × 60

but what is E? is it 1.5, or 2 oz?

 

[haruspex]

The only way I can make sense of the question is to suppose that ε and σ mean the same.

 

[mathguy2]

But if they're the same why are there different values ? That's the problem exactly as it's written, and I agree, it makes no sense.

Perhaps this information: "The United States Tennis Association imposes a rule on ball manufacturers:

σ = 1.5"

is a red herring? Cruel, but possible?

 

[haruspex]

But if they're the same why are there different values ?

The first is for a tennis ball, the second for a superball.

 

[mathguy2]

So then for: "If the racket is moving at 60 mph, how fast does the ball come off the racket?"

Vball = ε × Vracket

Vball = ε × 60

Vball = 1.5 x 60

Vball = 90mph if the racket moves at 60mph?

 

[haruspex]

So then for: "If the racket is moving at 60 mph, how fast does the ball come off the racket?"

Vball = ε × Vracket

Vball = ε × 60

Vball = 1.5 x 60

Vball = 90mph if the racket moves at 60mph?

I have no better suggestion. You said the other questions were easy. If the above is correct, does it seem about the right level of difficulty?

 

[mathguy2]

It seems too remedial, but when working through all three plug and chug solutions, the numbers seem to be accurate representations of what would likely occur. I think you're all correct, though. The wording of the problem is just ambiguous to a certain extent, and I think it lends itself to confusion and perhaps overthinking. The wording, "2 oz ball" early in the problem was confusing because I kept thinking I had to use that constant, but I think its just poor writing to a certain extent. The "2 oz" ball just refers to the entire problem, I guess, where the ball is always less than 2 oz's in every formula.

Solution 1: Vball = 90mph if the racket moves at 60mph.

Solution 2: Vracket = 80mph if the serve travels at 120 mph

Solution 3 (superball): Vball = 133 mph w/ superball and racket speed of 70mph

 

[mathguy2]

I appreciate the replies very much, though, because I would've sat here and squinted at this problem for a few hours (I wish that were hyperbole, but it isn't).