A field hockey player's arm and stick form a rigid segment. Assuming she hits the ball when the stick is vertical, which of the following would result in the greatest velocity of the end of her stick immediately before she hits the ball?
a) She stands still and swings her arm/stick at 300 deg/s
b) She runs forward at 4 m/s and swings her arm/stick at 300 deg/s
c)She stands still and swings her arm/stick at 150 deg/s
d) She runs forward at 2 m/s and swings her arm/stick at 150 deg/s
If you could explain why you chose your answer as well. Thanks :)
To determine which scenario would result in the greatest velocity of the end of the field hockey player's stick, we need to consider the combination of the angular velocity of the arm/stick and the linear velocity resulting from running forward.
Let's break down each scenario and calculate the overall velocity:
a) She stands still and swings her arm/stick at 300 deg/s:
In this scenario, the only movement contributing to the stick's velocity is the angular velocity of the arm/stick swinging. Here, the linear velocity is zero since she is standing still.
b) She runs forward at 4 m/s and swings her arm/stick at 300 deg/s:
Here, we have both angular and linear velocities. The linear velocity resulting from running forward, 4 m/s, is added to the angular velocity of the arm/stick swinging, 300 deg/s.
c) She stands still and swings her arm/stick at 150 deg/s:
Similar to scenario a), in this case, only the angular velocity of the arm/stick swinging contributes to the overall velocity.
d) She runs forward at 2 m/s and swings her arm/stick at 150 deg/s:
Similar to scenario b), this scenario combines both angular and linear velocities. The linear velocity resulting from running forward, 2 m/s, is added to the angular velocity of the arm/stick swinging, 150 deg/s.
Now, let's compare the combinations of angular and linear velocities in each scenario:
a) Angular velocity: 300 deg/s, Linear velocity: 0 m/s
b) Angular velocity: 300 deg/s, Linear velocity: 4 m/s
c) Angular velocity: 150 deg/s, Linear velocity: 0 m/s
d) Angular velocity: 150 deg/s, Linear velocity: 2 m/s
To calculate the overall velocity of the end of the player's stick, we can use a concept called vector addition. We can treat the angular and linear velocities as vectors and find their resultant vector to determine the overall velocity.
Calculating the magnitude of the resultant velocity using the Pythagorean theorem:
a) Resultant velocity: √((300^2) + (0^2)) = 300 units of velocity
b) Resultant velocity: √((300^2) + (4^2)) ≈ 300.04 units of velocity
c) Resultant velocity: √((150^2) + (0^2)) = 150 units of velocity
d) Resultant velocity: √((150^2) + (2^2)) ≈ 150.02 units of velocity
Comparing the resultant velocities, we can see that scenario b) - She runs forward at 4 m/s and swings her arm/stick at 300 deg/s - would result in the greatest velocity of the end of her stick immediately before she hits the ball, with an approximate velocity of 300.04 units.
Therefore, the correct answer is b).