A football kicker lines up to kick a game winning kick. The goalpost is 30m away and the ball must clear the 3.3m high cross-bar. If the kick leaves his foot at an angle of 35 degrees, how fast must the kick be to clear the bar?
To calculate the speed needed for the football to clear the bar, we can use the principles of projectile motion. The horizontal and vertical components of the ball's initial velocity can be determined based on the angle at which it leaves the kicker's foot.
Let's break down the problem into its components:
1. Horizontal Motion:
Since there are no external horizontal forces acting on the football, the horizontal velocity remains constant throughout the kick. The horizontal distance, in this case, is the distance to the goalpost, which is 30m.
2. Vertical Motion:
In the vertical direction, the football will experience gravitational acceleration, which is approximately 9.8 m/s^2. Initially, the ball is kicked at an angle, so we need to consider both the vertical and horizontal components of its velocity.
Now, let's calculate the vertical component of the initial velocity using trigonometry. The formula for the vertical component (Vv) is:
Vv = V * sin θ
Where:
Vv is the vertical component of the initial velocity
V is the initial velocity of the kick
θ is the angle at which the kick is made (35 degrees in this case)
Next, let's calculate the time it takes for the ball to reach its maximum height using the formula:
t = Vv / g
Where:
t is the time
g is the acceleration due to gravity (9.8 m/s^2)
The time to reach the maximum height is the same as the time it takes to fall from the maximum height to the cross-bar level.
Then, we can calculate the time it takes for the ball to reach the cross-bar level from the maximum height using the formula:
t_total = 2 * t
Now, we can determine the vertical distance the ball has to travel using the following equation:
H = Vv * t_total + (1/2) * g * t_total^2
Where:
H is the total vertical distance traveled by the ball (3.3 m in this case)
Finally, by rearranging the formula, we can solve for the initial velocity (V) of the kick:
V = H / (sin θ * t_total)
Using the values given, you can substitute them into the equations to find the required speed for the kick to clear the bar.
Vo^2*sin(2A)/g = Dx = 30 m.
Vo^2*sin(70) / 9.8 = 30,
Vo^2*0.095887 = 30,
Vo^2 = 30 / 0.095887 = 312.9,
Vo = 17.69 m/s.