A place kicker must kick a football from a point 35.0 m (= 38.3 yd) from the goal, and the ball must clear the crossbar, which is 3.05 m high. When kicked, the ball leaves the ground with a speed of 19.8 m/s at an angle of 45.0o to the horizontal. By how much does the ball clear the crossbar (if in fact it does)? Enter positive values if the ball clears and negative values if it falls below the crossbar. (in m)
What is the vertical velocity of the ball at the time it reaches the crossbar? Enter positive values if it is still rising and negative values if it is falling. (in m/s)
To find out how much the ball clears or falls below the crossbar, we need to calculate the maximum height reached by the ball.
The initial speed of the ball can be resolved into horizontal and vertical components:
Vx = V * cos(θ) = 19.8 m/s * cos(45°) ≈ 13.99 m/s
Vy = V * sin(θ) = 19.8 m/s * sin(45°) ≈ 13.99 m/s
The time it takes for the ball to reach the maximum height can be found using the vertical component of velocity:
Vy = Vy0 - g * t
0 = 13.99 m/s - 9.8 m/s^2 * t
t = 13.99 m/s / 9.8 m/s^2 ≈ 1.43 s
To find the maximum height, we can use the equation:
y = y0 + Vyo * t - 0.5 * g * t^2
y = 0 + 13.99 m/s * 1.43 s - 0.5 * 9.8 m/s^2 * (1.43 s)^2
y ≈ 10.01 m
The ball's maximum height is approximately 10.01 m.
Now we can calculate how much the ball clears or falls below the crossbar:
Clearance = maximum height - crossbar height
Clearance = 10.01 m - 3.05 m
Clearance ≈ 6.96 m
Therefore, the ball clears the crossbar by approximately 6.96 m.
To determine the vertical velocity of the ball at the time it reaches the crossbar, we need to find the vertical component of velocity at that time.
Using the equation:
Vy = Vy0 - g * t
Vy = 13.99 m/s - 9.8 m/s^2 * 1.43 s
Vy ≈ -0.98 m/s
The vertical velocity of the ball at the time it reaches the crossbar is approximately -0.98 m/s, indicating that it is falling.
To find out how much the ball clears or falls below the crossbar and the vertical velocity of the ball when it reaches the crossbar, we need to analyze the motion of the ball.
Step 1: Break down the initial velocity of the ball into its horizontal and vertical components.
The initial velocity can be separated into two components:
- The horizontal component: v₀x = v₀ * cos(θ)
- The vertical component: v₀y = v₀ * sin(θ)
where v₀ is the initial velocity (19.8 m/s) and θ is the angle (45.0o).
Step 2: Determine the time it takes for the ball to reach the crossbar using the vertical motion.
Using the equation: Δy = v₀y * t + (1/2) * g * t²
where Δy is the vertical displacement (difference in height between the crossbar and the initial position), v₀y is the initial vertical velocity, g is the acceleration due to gravity (9.8 m/s²), and t is the time.
In this case, Δy is the height of the crossbar (3.05 m) and we want to solve for t.
Step 3: Solve for t using the quadratic equation.
The equation becomes: 3.05 = (19.8 * sin(45.0)) * t + (1/2) * 9.8 * t²
Simplifying the equation: 3.05 = 13.959 t + 4.9 t²
Rearranging terms: 4.9 t² + 13.959 t - 3.05 = 0
We can solve this quadratic equation to find the time it takes for the ball to reach the crossbar.
Step 4: Calculate the time the ball takes to reach the crossbar.
Solving the quadratic equation yields two possible values for t. Since the ball is still rising at the time it reaches the crossbar, we can reject the negative value.
Step 5: Calculate how much the ball clears or falls below the crossbar.
Using the equation: Δy = v₀y * t + (1/2) * g * t²
Substitute the known values:
Δy = (19.8 * sin(45.0)) * t + (1/2) * 9.8 * t²
Evaluate Δy to find out how much the ball clears or falls below the crossbar.
Step 6: Calculate the vertical velocity of the ball when it reaches the crossbar.
Using the equation: v = v₀y + g * t
Substitute the known values:
v = (19.8 * sin(45.0)) + 9.8 * t
Evaluate v to determine if the ball is still rising or falling when it reaches the crossbar.
By following these steps, you should be able to find the answers to the given question.
The horizontal and vertical speed are both 19.8/√2 = 14 m/s
At that speed, it takes 35/14 = 2.5 seconds to get to the goal.
The height of the ball is
y = 14t-4.9t^2
y(2.5) = 4.375 m
Looks like it will clear the bar by 1.325 m
I'll let you decide the vertical velocity at t=2.5
v = 14-9.8t