Final answer:
To find the initial speed of a soccer ball that must pass over a goal 2.4 m high from a 30 m distance, we analyze the projectile motion into horizontal and vertical components, considering the ball's angle of 40° and the acceleration due to gravity.
Step-by-step explanation:
Finding the Initial Speed of a Soccer Ball
To find the initial speed of a soccer ball kicked at a 40° angle to just pass over the goal, we can use the physics of projectile motion.
The goal is 2.4 m above the ground and the ball is kicked from a distance of 30 m.
The trajectory of a projectile is determined by its initial speed, the angle of projection, and the acceleration due to gravity.
We can separate the problem into horizontal and vertical components.
First, we determine how long it takes for the ball to travel 30 m horizontally.
Then, using this time, we find the initial vertical speed that will let the ball reach a height of 2.4 m.
The horizontal component of initial speed (vix) is vi × cos(40°).
The vertical component of initial speed (viy) is vi × sin(40°).
Using the equation for horizontal motion, s = vix × t, where s is the horizontal distance (30 m), we can solve for time (t).
Then, for vertical motion, we use the equation s = viy × t + 0.5 × g × t2, where g is the acceleration due to gravity (-9.8 m/s2) and the displacement s is 2.4 m.
Solving these equations simultaneously gives us the initial speed vi.
A ball Is thrown vertically upward with a speed of 40.0 m/s. How high does It rise? How long does it take to reach its highest point? How long does the ball take to hit the ground after it reaches its highest point? What is its velocity when it returns to the level from which it started?