A place kicker must kick the football from a point 34.2 m from the goal and
clear a bar 3.00 m above the ground. The ball leaves the ground with a speed of 20.0 m/s at an angle of 53.0 degrees above the horizontal.
(a) By how much (m) does the ball clear (positive value) or fall short (negative value) of the cross bar? (This is vertical distance above or below the cross bar.)
(b) When it gets to the cross bar, what is the vertical component of the ball's velocity (m/s)? (Is it rising or falling-pay attention to the sign?)
(a) The horizontal velocity component is 20 cos 53 = 12.036 m/s. To travel 34.2 m to the goal post requires 34.2/12.036 = 2.84 seconds. Its height y at that time is
y(t=2.84) = 20*sin53*t - (g/2)t^2
= 5.84 m. That would be 2.84 m above the cross bar.
(2) V(t=2.84) = 20sin53 - g*t = 15.97 - 27.83 = -11.86 m/s. So the football is falling
To solve this problem, we can use the kinematic equations to analyze the horizontal and vertical components of the motion separately. Let's break down the problem step-by-step:
Step 1: Determine the initial vertical and horizontal velocities
The ball leaves the ground with a speed of 20.0 m/s. We can find the vertical and horizontal components of this initial velocity using trigonometry:
Vertical component: Vy = V * sin(θ) = 20.0 * sin(53.0°)
Horizontal component: Vx = V * cos(θ) = 20.0 * cos(53.0°)
Step 2: Calculate the time of flight
Using the vertical component of velocity, we can determine the time it takes for the ball to reach its maximum height and then return to the ground:
Vy = V0y + g * t
0 = Vy - 9.8 * t
Solve for t:
t = Vy / 9.8
Step 3: Calculate the maximum height above the ground
To find the maximum height, we can substitute the time obtained in Step 2 into the vertical displacement equation:
hmax = V0y * t - 0.5 * g * t^2
Step 4: Determine the vertical distance above or below the crossbar
The crossbar is 3.00 m above the ground. We can calculate how much the ball clears or falls short of the crossbar by subtracting the maximum height from the height of the crossbar:
vertical distance = hmax - 3.00
Step 5: Find the vertical component of the velocity at the crossbar
Since we know the time of flight, we can calculate the vertical component of the velocity at that moment using:
Vy = V0y - g * t
Now, let's plug in the numbers and solve the problem:
Step 1:
Vy = 20.0 * sin(53.0°)
Vy ≈ 16.20 m/s
Vx = 20.0 * cos(53.0°)
Vx ≈ 11.18 m/s
Step 2:
t = 16.20 / 9.8
t ≈ 1.65 s
Step 3:
hmax = 16.20 * 1.65 - 0.5 * 9.8 * (1.65)^2
hmax ≈ 14.00 m
Step 4:
vertical distance = 14.00 - 3.00
vertical distance ≈ 11.00 m
The ball clears the crossbar by approximately 11.00 meters.
Step 5:
Vy = 16.20 - 9.8 * 1.65
Vy ≈ 0.42 m/s
The vertical component of the velocity at the crossbar is approximately 0.42 m/s, which means the ball is falling (negative sign implies direction).
To solve this problem, we can break down the motion of the ball into horizontal and vertical components. Let's tackle each part one by one.
(a) To determine how much the ball clears or falls short of the crossbar, we need to find the vertical displacement of the ball. We can use the equation:
y = y₀ + v₀y * t - (1/2) * g * t²
where:
y is the vertical displacement
y₀ is the initial vertical position (height above the ground)
v₀y is the initial vertical component of velocity
t is the time of flight
g is the acceleration due to gravity (approximately 9.8 m/s²)
First, let's find the time of flight (t). We can use the initial vertical velocity (v₀y) and the acceleration due to gravity (g) in the equation:
v₀y = v₀ * sin(θ)
where:
v₀ is the initial velocity (20.0 m/s)
θ is the angle of projection (53.0 degrees)
v₀y = 20.0 m/s * sin(53.0 degrees)
≈ 15.303 m/s
Next, let's find the time of flight (t) using the equation:
t = 2 * v₀y / g
t = 2 * 15.303 m/s / 9.8 m/s²
≈ 3.12 s
Now, we can substitute the values into the displacement equation:
y = 0 + (15.303 m/s) * (3.12 s) - (1/2) * (9.8 m/s²) * (3.12 s)²
≈ 47.99 m
The vertical displacement of the ball is approximately 47.99 m. Since the target bar is at a height of 3.00 m, the ball clears the crossbar by:
47.99 m - 3.00 m
≈ 44.99 m
Therefore, the ball clears the crossbar by approximately 44.99 m.
(b) We can determine the vertical component of the ball's velocity when it reaches the crossbar. Using the equation:
v_y = v₀y - g * t
where:
v_y is the vertical component of velocity
v₀y is the initial vertical component of velocity
t is the time of flight (already calculated as 3.12 s)
Substituting the values:
v_y = 15.303 m/s - (9.8 m/s²) * (3.12 s)
≈ -19.436 m/s
The negative sign indicates that the vertical velocity component is directed downwards or falling. Therefore, when the ball reaches the crossbar, its vertical component of velocity is approximately -19.436 m/s.