A soccer player kicks the ball toward a goal that is 15.0 m in front of him. The ball leaves his foot at a speed of 17.1 m/s and an angle of 35.0 ° above the ground. Find the speed of the ball when the goalie catches it in front of the net.
To find the speed of the ball when the goalie catches it, we need to break down the initial velocity of the ball into its horizontal and vertical components.
The horizontal component (Vx) of the ball's initial velocity remains constant throughout the motion because there is no horizontal acceleration.
The vertical component (Vy) of the initial velocity will change due to gravity. At the highest point of the ball's trajectory, the vertical component of velocity becomes zero.
We can calculate the time (t) it takes for the ball to reach the highest point using the vertical component of the initial velocity:
Vy = Voy - gt
Where:
- Vy is the vertical component of the initial velocity
- Voy is the initial vertical velocity (Voy = Vosinθ)
- g is the acceleration due to gravity (approximately 9.8 m/s²)
In this case, the initial vertical velocity (Voy) is Vosinθ, where Vo is the initial speed of the ball (17.1 m/s) and θ is the launch angle (35.0°). Therefore:
Voy = Vo * sin(θ)
Next, we can solve for t using the equation:
0 = Voy - gt
Solving for t, we get:
t = Voy / g
Now, we can calculate the time it takes for the ball to reach the highest point. Let's substitute the values into the equation:
t = (Vosinθ) / g
Once we have found the time it takes for the ball to reach the highest point, we can determine the total time it takes for the ball to reach the goalie by doubling the time it took to reach the highest point:
total time = 2 * t
Now, we can determine how far the ball travels horizontally by using the horizontal component of the initial velocity:
Sx = Vxt
Substituting the values:
Sx = Vx * (2t)
To find the horizontal component of the initial velocity (Vx), we can use the equation:
Vx = Vocosθ
Substituting the values:
Vx = Vo * cos(θ)
Finally, we calculate the magnitude of the ball's final velocity (v) using the Pythagorean theorem:
v = √(Sx² + Sy²)
Substituting the values:
v = √(Sx² + (15.0 m)²)
Now, let's calculate the solution step by step:
Vo = 17.1 m/s [Given]
θ = 35.0° [Given]
g = 9.8 m/s² [Acceleration due to gravity]
Voy = Vo * sin(θ)
t = (Vosinθ) / g
total time = 2 * t
Vx = Vo * cos(θ)
Sx = Vx * (2t)
v = √(Sx² + (15.0 m)²)
By systematically solving these equations, we can find the speed of the ball when the goalie catches it in front of the net.