A football player kicks a ball with a soeed of 22.4 m/s at an angle of 49 degrees above the horizontal from a distance of 39.0 m from the goal post,
A). By how much does the ball clear or fall short of clearing the crossbar of the goalpost if the that bar is 3.05 m high?
B). What is the vetical velocity of the ball at the time it reaches the goalpost?
previous questions.. It's a confusing process for me to put what numbers where and use equations.
To answer these questions, we can break down the motion of the ball into its horizontal and vertical components.
Let's start by finding the time it takes for the ball to reach the goalpost. We can use the horizontal component of the velocity for this.
Step 1: Find the horizontal component of the initial velocity:
The horizontal velocity can be found using the equation: Vx = V * cos(theta), where V is the initial velocity and theta is the angle above the horizontal.
Vx = 22.4 m/s * cos(49 degrees) = 14.17 m/s
Step 2: Find the time of flight:
The time of flight can be calculated using the equation: t = d / Vx, where d is the horizontal distance traveled.
t = 39.0 m / 14.17 m/s = 2.76 s
A) Now let's calculate by how much the ball clears or falls short of clearing the crossbar of the goalpost.
To find the vertical position of the ball at this time, we can use the vertical component of the initial velocity.
Step 3: Find the vertical component of the initial velocity:
The vertical velocity can be found using the equation: Vy = V * sin(theta), where V is the initial velocity and theta is the angle above the horizontal.
Vy = 22.4 m/s * sin(49 degrees) = 16.05 m/s
Step 4: Find the vertical displacement:
The vertical displacement can be found using the equation: h = Vy * t + (1/2) * g * t^2, where h is the vertical displacement, Vy is the vertical velocity, t is the time of flight, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
h = 16.05 m/s * 2.76 s + (1/2) * 9.8 m/s^2 * (2.76 s)^2 = 44.14 m
The crossbar of the goalpost is 3.05 m high, so the ball clears the crossbar by:
Clearance = h - crossbar height = 44.14 m - 3.05 m = 41.09 m
Therefore, the ball clears the crossbar by 41.09 meters.
B) Now let's calculate the vertical velocity of the ball when it reaches the goalpost.
The vertical velocity at any given time can be found using the equation: Vy = V * sin(theta) - g * t, where V is the initial velocity, theta is the angle above the horizontal, g is the acceleration due to gravity, and t is the time of flight.
Vertical velocity at the time the ball reaches the goalpost is:
Vy = 22.4 m/s * sin(49 degrees) - 9.8 m/s^2 * 2.76 s = 16.05 m/s - 27.048 m/s = -10.998 m/s
The negative sign indicates that the velocity at that time is in the opposite direction of the initial velocity. Therefore, the vertical velocity of the ball at the time it reaches the goalpost is 10.998 m/s downward.
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