Final answer:
The initial speed of the soccer ball is 13 m/s, calculated using the Pythagorean theorem. It will land approximately 12.24 m away, and at its maximum height, the ball will be moving at a horizontal speed of 12 m/s.
Step-by-step explanation:
To determine the initial speed of the soccer ball as it is kicked, we need to use the Pythagorean theorem since we have both a horizontal and a vertical component of the initial velocity. The initial speed of the ball can be calculated as:
Initial speed = √(horizontal velocity)^2 + (vertical velocity)^2
= √(12 m/s)^2 + (5 m/s)^2
= √144 + 25
= √169
= 13 m/s.
This is the initial speed of the ball as it leaves the kicker's foot.
To find out how far away the ball will land, the horizontal velocity (which remains constant throughout the flight since we are ignoring air resistance) and the time the ball is in the air are important. Given that the vertical velocity will determine the time in the air, we can use the equation for the time of flight, which is derived from the vertical motion:
Time of flight = 2 × (vertical velocity) / g
= 2 × 5 m/s / 9.81 m/s^2
≈ 1.02 s,
where g is the acceleration due to gravity (9.81 m/s^2), and we have multiplied by 2 to account for both the ascent and descent of the ball.
With the time in the air and the constant horizontal velocity, we can find the distance it will travel as:
Distance = horizontal velocity × time of flight
= 12 m/s × 1.02 s
≈ 12.24 m.
This is the distance where the ball will land back on the ground.
To determine how fast the soccer ball will be going just as it reaches maximum height, we only consider the horizontal component of the velocity as the vertical component will be zero at the maximum height. Thus, the ball will be moving at a constant speed of 12 m/s horizontally even at maximum height.