At an instant when a soccer ball is in contact with the foot of a player kicking it, the horizontal or x component of the ball’s acceleration is 893 m/s2 and the vertical or y component of its acceleration is 1,114 m/s2. 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 calculate the magnitude of the net force acting on the soccer ball at the instant it is in contact with the player’s foot, we can use Newton’s Second Law of Motion, which states:F=maF = maF=ma
Where:
- F is the net force,
- m is the mass of the ball,
- a is the acceleration of the ball.
Since the acceleration is given in both the horizontal (x) and vertical (y) directions, we need to find the resultant (or net) acceleration, which can be done by using the Pythagorean theorem:anet=ax2+ay2a_{\text{net}} = \sqrt{a_x^2 + a_y^2}anet=ax2+ay2
Where:
- ax=893 m/s2a_x = 893 \, \text{m/s}^2ax=893m/s2 is the horizontal acceleration,
- ay=1,114 m/s2a_y = 1,114 \, \text{m/s}^2ay=1,114m/s2 is the vertical acceleration.
Substitute the values:anet=(893)2+(1114)2a_{\text{net}} = \sqrt{(893)^2 + (1114)^2}anet=(893)2+(1114)2anet=797,649+1,242,196a_{\text{net}} = \sqrt{797,649 + 1,242,196}anet=797,649+1,242,196anet=2,039,845a_{\text{net}} = \sqrt{2,039,845}anet=2,039,845anet≈1,427.9 m/s2a_{\text{net}} \approx 1,427.9 \, \text{m/s}^2anet≈1,427.9m/s2
Now, we can calculate the net force using the mass of the ball m=0.45 kgm = 0.45 \, \text{kg}m=0.45kg:F=(0.45)×(1,427.9)F = (0.45) \times (1,427.9)F=(0.45)×(1,427.9)F≈643.6 NF \approx 643.6 \, \text{N}F≈643.6N
Therefore, the magnitude of the net force acting on the soccer ball is approximately 643.6 Newtons.
Explanation:
The force required to accelerate the soccer ball is the result of both horizontal and vertical accelerations. Since the accelerations are in perpendicular directions (horizontal and vertical), they are combined using the Pythagorean theorem to get the resultant acceleration. Once the net acceleration is found, the force is calculated using Newton’s Second Law. The magnitude of the force is quite large because of the high acceleration components in both directions.
