What Is the Impulse Imparted to a Soccer Ball When 'Heading' It?

[Becca93]

Homework Statement

Recent studies have raised concern about 'heading' in youth soccer (i.e., hitting the ball with the head). A soccer player 'heads' a size 5 ball deflecting it by 31.0°, and keeps its speed of 10.40 m/s constant. A size 5 ball has a mass of approximately 0.446 kg. What is the magnitude of the impulse which the player must impart to the ball?

After getting this incorrect, I was given the hint: "The impulse is the momentum imparted to the ball, and it changes the ball's momentum. Remember that momentum is a vector."The attempt at a solution

My prof has done this question in class a number of times, however the answers he came to in class are not being accepted as correct answers.

The way we did it in class is as follows:

I = fΔt = Δp
I = (Ʃp)f - (Ʃp)i
I = (Ʃmv)f - (Ʃmv)i
I = m(mf-mi) = mΔv

"Since the velocity magnitude is the same before and after collision, only the change in direction contributes to the change in velocity."

Δv = v

10.4cos31 = 8.91 m/s

Original classroom answer:

I = (.446)(8.91)
I = 30972 N*s

Revised classroom answer:

|Δv| = (8.9 - 10.4)

I = Δp = mΔv = m(v2 - v1) = m(10.4mgcosθ - 10.4)
I = m(8.9 - 10.4)
I = -0.669As I said, both of these answers are incorrect. Can anyone help explain what is wrong and what process needs to be followed to get the right answer? The question feels like it should be simple, but I just can't seem to spot what where my prof went wrong explaining it to the class.

 

[Doc Al]

You must find the change in velocity by subtracting the velocity vectors. Take the original velocity direction as the +x axis. Now find the components of the final velocity vector. Subtract to find the components of the change in velocity vector, then find its magnitude. (You can subtract the vectors any way you like; using components is just one way.)

 

[I like Serena]

Hi Becca93!

Since your speed before equals your speed after, the corresponding vectors span an isosceles triangle with an angle of 31 degrees in between.
Do you know how to calculate the length of the third side?

 

[Becca93]

You must find the change in velocity by subtracting the velocity vectors. Take the original velocity direction as the +x axis. Now find the components of the final velocity vector. Subtract to find the components of the change in velocity vector, then find its magnitude. (You can subtract the vectors any way you like; using components is just one way.)

Okay, I think I understand how to do that now. My prof and I must have misunderstood the question. Thank you!

Hi Becca93!

Since your speed before equals your speed after, the corresponding vectors span an isosceles triangle with an angle of 31 degrees in between.
Do you know how to calculate the length of the third side?

Got it! Thanks! I don't know why it didn't register to me that since it was an isosceles triangle I could use another method to find the base. Thank you.

 

[asteriatic]

There are a few issues with the approach used in class. First, the impulse is not equal to the change in momentum, but rather the integral of the force over time. In this case, the impulse is equal to the change in momentum, but this may not always be the case.

Second, in the original classroom answer, the units for impulse are incorrect. The units for impulse are kg*m/s, not N*s. The units for the revised classroom answer are correct, but the answer is negative, which does not make sense for an impulse.

To correctly solve this problem, we can use the formula I = FΔt = Δp, where F is the average force applied to the ball, Δt is the time over which the force is applied, and Δp is the change in momentum of the ball.

First, we need to find the average force applied to the ball. We can use Newton's second law, F = ma, to find this. The acceleration of the ball is given by a = v^2/R, where v is the speed of the ball and R is the radius of the ball. We can use the given information to find the radius of the ball, R = 0.11 m.

Next, we can find the average force by plugging in the values for mass, acceleration, and time into the formula F = ma. We know that the time over which the force is applied is the same as the time it takes for the ball to change direction, which is equal to the time it takes for the ball to travel half of its circumference, t = πR/v.

So, F = (0.446 kg)(10.4^2/0.11 m)/(π(0.11 m)/10.4 m/s) = 309.72 N.

Finally, we can use the formula I = FΔt = Δp to find the impulse. We know that the impulse is equal to the change in momentum, which is equal to the initial momentum minus the final momentum.

Δp = (0.446 kg)(10.4 m/s) - (0.446 kg)(10.4 m/s) = 0 kg*m/s

Therefore, the impulse is equal to the average force times the time over which it is applied, I = FΔt = (309.72 N)(π(0.11 m)/10