A place kicker must kick a football from a point 29.0 m (= 31.7 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 17.8 m/s at an angle of 41.0o to the horizontal. By how much does the ball clear the crossbar (if in fact it does)? What is the vertical velocity of the ball at the time it reaches the crossbar?
To find out by how much the ball clears the crossbar, we first need to determine the range of the football, which is the horizontal distance it travels.
The range of a projectile can be calculated using the formula:
Range = (initial velocity)^2 * sin(2θ) / g
where:
- Initial velocity is the launch speed of the ball.
- θ is the launch angle.
- g is the acceleration due to gravity (approximately 9.8 m/s^2).
Let's calculate the range:
Range = (17.8 m/s)^2 * sin(2 * 41.0°) / 9.8 m/s^2
Using a scientific calculator or software, we find that the range is approximately 37.2 meters.
Since the goal is 29.0 meters away, we can conclude that the ball will easily clear the crossbar.
Next, let's find the vertical velocity of the ball when it reaches the crossbar. The vertical velocity at any given time during the projectile motion can be found using the formula:
Vertical velocity = initial velocity * sin(θ) - g * t
where:
- Initial velocity is the launch speed of the ball.
- θ is the launch angle.
- g is the acceleration due to gravity.
- t is the time at which we want to find the vertical velocity.
To determine the time it takes for the ball to reach the crossbar, we can use the fact that the horizontally projected motion has no initial vertical velocity. This means the time taken to reach the crossbar is the same as the time taken to reach the highest point of the trajectory.
The time to reach the maximum height can be calculated using the formula:
Time = initial vertical velocity / g
Since the initial vertical velocity is given by:
Initial vertical velocity = initial velocity * sin(θ)
Let's find the time taken to reach the maximum height:
Time = (17.8 m/s * sin(41.0°)) / 9.8 m/s^2
Using a scientific calculator or software, we find that the time taken to reach the maximum height is approximately 1.32 seconds.
Now, to find the vertical velocity when the ball reaches the crossbar, we use the time calculated above in the formula:
Vertical velocity = initial velocity * sin(θ) - g * t
Vertical velocity = 17.8 m/s * sin(41.0°) - 9.8 m/s^2 * 1.32 s
Using a calculator, we find that the vertical velocity when the ball reaches the crossbar is approximately 7.9 m/s.
Therefore, the ball clears the crossbar by approximately (3.05 m - 7.9 m/s) = -4.85 m, which means it falls short of clearing the crossbar by 4.85 meters.