At an instant when soccer ball is in contact with the foot of a player kicking it, the horizontal or x component of the ball’s acceleration is 512 m/s? and the vertical or y component of its acceleration is 1400 m/s? . The ball’s mass is 0.45 kg: What is the magnitude of the net force acting on the soccer ball at this instant?
The Correct Answer and Explanation is:
To find the magnitude of the net force acting on the soccer ball, we can use Newton’s second law, which states that:F=maF = maF=ma
Where:
- FFF is the net force,
- mmm is the mass of the object (in this case, the soccer ball), and
- aaa is the acceleration.
Step 1: Find the total acceleration
The components of the acceleration in the horizontal (x) and vertical (y) directions are given as:
- ax=512 m/s2a_x = 512 \, \text{m/s}^2ax=512m/s2
- ay=1400 m/s2a_y = 1400 \, \text{m/s}^2ay=1400m/s2
To find the total (or net) acceleration aaa, we combine these two components using the Pythagorean theorem, since the x and y directions are perpendicular to each other:a=ax2+ay2a = \sqrt{a_x^2 + a_y^2}a=ax2+ay2
Substituting the given values:a=(512)2+(1400)2a = \sqrt{(512)^2 + (1400)^2}a=(512)2+(1400)2a=262,144+1,960,000a = \sqrt{262,144 + 1,960,000}a=262,144+1,960,000a=2,222,144a = \sqrt{2,222,144}a=2,222,144a≈1491.4 m/s2a \approx 1491.4 \, \text{m/s}^2a≈1491.4m/s2
Step 2: Apply Newton’s second law
Now that we have the total acceleration, we can apply Newton’s second law to find the net force. The mass of the soccer ball is given as:m=0.45 kgm = 0.45 \, \text{kg}m=0.45kg
Using the formula F=maF = maF=ma:F=(0.45 kg)×(1491.4 m/s2)F = (0.45 \, \text{kg}) \times (1491.4 \, \text{m/s}^2)F=(0.45kg)×(1491.4m/s2)F≈671.13 NF \approx 671.13 \, \text{N}F≈671.13N
Final Answer:
The magnitude of the net force acting on the soccer ball at this instant is approximately 671.13 N.
Explanation:
The acceleration components in both the x and y directions were given. To find the total acceleration, we used the Pythagorean theorem because the accelerations are acting in perpendicular directions. After finding the net acceleration, we applied Newton’s second law to calculate the net force, considering the ball’s mass and the total acceleration it experiences. This method applies the principles of vector addition and Newton’s second law of motion.
