At an instant when a soccer ball is in contact with the foot of the player kicking it, the horizontal or x component of the ball’s acceleration is 810 m/s² and the vertical or y component of its acceleration is 760 m/s². The ball’s mass is 0.42 kg. What is the magnitude of the net force acting on the soccer ball at this instant? Number i Save for Later Units Attempts: 0 of 1 used Submit Answer
The Correct Answer and Explanation is :
To calculate the magnitude of the net force acting on the soccer ball, we need to use Newton’s second law of motion, which states that:
[
F_{\text{net}} = m \cdot a
]
where:
- ( F_{\text{net}} ) is the net force acting on the object,
- ( m ) is the mass of the object (in this case, the soccer ball),
- ( a ) is the acceleration of the object.
Step 1: Find the acceleration components
We are given the horizontal (x) and vertical (y) components of acceleration:
- ( a_x = 810 \, \text{m/s}^2 ),
- ( a_y = 760 \, \text{m/s}^2 ).
Step 2: Calculate the resultant acceleration
The resultant (or total) acceleration ( a ) can be found by combining the horizontal and vertical components using the Pythagorean theorem:
[
a = \sqrt{a_x^2 + a_y^2}
]
Substituting the given values:
[
a = \sqrt{(810)^2 + (760)^2}
]
[
a = \sqrt{656100 + 577600}
]
[
a = \sqrt{1235700}
]
[
a \approx 1111.2 \, \text{m/s}^2
]
Step 3: Calculate the net force
Now that we have the magnitude of the acceleration, we can use Newton’s second law to find the net force. The mass of the soccer ball is given as 0.42 kg, so:
[
F_{\text{net}} = m \cdot a
]
[
F_{\text{net}} = 0.42 \, \text{kg} \cdot 1111.2 \, \text{m/s}^2
]
[
F_{\text{net}} \approx 466.7 \, \text{N}
]
Conclusion:
The magnitude of the net force acting on the soccer ball at this instant is approximately 467 N.
This calculation assumes that the horizontal and vertical components of acceleration are independent of each other and that the ball is being acted upon by an external force (the player’s foot) that results in these accelerations. The net force is the combination of these two components, which is why we used the Pythagorean theorem to find the total acceleration.