Suppose a soccer player kicks the ball from a distance 30 m toward the goal. Find the initial speed of the ball if it just passes over the goal, 2.4 m above the ground, given the initial direction to be above the horizontal.
40 degrees*
10.67 m/s
Suppose a soccer player kicks the ball from a distance 15 m toward the goal. Find the initial speed of the ball if it just passes over the goal, 2.4 m above the ground, given the initial direction to be 35° above the horizontal.
To find the initial speed of the ball, we can use the kinematic equation that relates the vertical displacement, initial velocity, and time of flight for an object in projectile motion.
Let's break down the problem step by step:
Step 1: Define the known variables:
- Distance of the ball from the player to the goal: 30 m
- Height of the goal above the ground: 2.4 m
- Acceleration due to gravity: 9.8 m/s²
Step 2: Determine the time of flight:
Since the ball passes over the goal, the vertical displacement is equal to the height of the goal. So, we can write the equation as follows:
Δy = (V₀y × t) + (0.5 × (-9.8) × t²)
Here, Δy is the vertical displacement (2.4 m) and V₀y is the vertical component of the initial velocity. Since the initial direction is above the horizontal, V₀y is positive.
Step 3: Calculate the vertical component of the initial velocity:
The initial velocity has components in both the horizontal and vertical directions. Since the ball just passes over the goal, the vertical component of the velocity at the top of the trajectory is zero. We can use this information to determine the vertical component of the initial velocity (V₀y).
Using the formula: Vfy = V₀y + (-9.8) × t, where Vfy is the final vertical velocity (0 m/s), we can find t.
0 = V₀y - 9.8 × t
Step 4: Substitute the value of t into our equation to find V₀y:
0 = V₀y - 9.8 × (Δy / V₀y)
0 = V₀y² - 9.8 × (2.4 / V₀y)
V₀y² = 23.52
V₀y = √(23.52)
Step 5: Calculate the total initial velocity:
Since the horizontal and vertical components of the initial velocity are perpendicular to each other, we can use the Pythagorean theorem to find the magnitude of the initial velocity (V₀).
Using the formula: V₀² = V₀x² + V₀y², where V₀x is the horizontal component of the initial velocity, and V₀y is the vertical component of the initial velocity, we need to find V₀x.
Given that the ball is kicked horizontally, V₀x is the same as the horizontal distance from the player to the goal (30 m).
V₀² = (30)² + (√(23.52))²
Step 6: Calculate the square root to find V₀:
V₀ = √(30² + √(23.52)²)
By evaluating this expression, you can find the initial speed of the ball.
[v sin40]^2 = 2 (9.8) 2.4. This assumes vy final is zero as it crosses the goal.
v = sqrt(2*9.8*2.4)/sin40