Velocity of football kick problem

[Phy.Student]

A football player kicks a field goal from a distance of 45m from the goalpost.
The football is launched at 35 degrees above horizontal.
What initial velocity is required so that the football just clears the goalpost crossbar that is 3.1m above the ground? ignore air resistance.




This question REALLY got me stomped because its considered a lvl 3 question (the hardest there is in this book) and I really have no idea where to begin this...

 

[berkeman]

A football player kicks a field goal from a distance of 45m from the goalpost.
The football is launched at 35 degrees above horizontal.
What initial velocity is required so that the football just clears the goalpost crossbar that is 3.1m above the ground? ignore air resistance.




This question REALLY got me stomped because its considered a lvl 3 question (the hardest there is in this book) and I really have no idea where to begin this...

Welcome to the PF.

What are the Relevant Equations? Write down the kinematic equations for constant acceleration -- those are your starting point.

Then draw a sketch of the geometry of the situation. The ball follows a parabolic arc, no? Its horizontal velocity is constant, and its vertical velocity as a function of time comes from those kinematic equations...

 

[Phy.Student]

Welcome to the PF.

What are the Relevant Equations? Write down the kinematic equations for constant acceleration -- those are your starting point.

Then draw a sketch of the geometry of the situation. The ball follows a parabolic arc, no? Its horizontal velocity is constant, and its vertical velocity as a function of time comes from those kinematic equations...

Well i copied everything that was on that question and the equations that we have been mostly using are
Vf= Vi + a t

Vf^2 = Vi^2 + 2 a d

D = Vi(t) + 1/2 (a) (t^2)

and I'm assuming it would be an arc since its nearly impossible for a football to travel in an uniform line.

 

[berkeman]

Well i copied everything that was on that question and the equations that we have been mostly using are
Vf= Vi + a t

Vf^2 = Vi^2 + 2 a d

D = Vi(t) + 1/2 (a) (t^2)

That's not quite all that you need, but is close. The ball starts from the ground and ends up 3.1m high at the end of its parabolic arc. So you have the initial y and final y. And it travels 45 meters horizontally, so you have your initial x and final x. You know the launch angle, so that gives you the vertical and horrizontal starting velocities, in terms of the unknown total velocity. The horizontal velocity is constant (why?), and the vertical velocity varies as you have shown in your equation.

Write two equations and solve for the two unknowns (probably initial overall velocity and the flight time...).

 

[rwisz]

Firstly, we can use the kinematic equations of motion to solve this problem. The initial velocity, or the velocity at which the football is launched, can be calculated using the equation: v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration (in this case, due to gravity) and t is the time taken.

In this problem, we know the distance (45m) and the angle of launch (35 degrees), but we need to find the initial velocity. To do this, we can break the initial velocity into its horizontal and vertical components. The horizontal component will remain constant throughout the motion, while the vertical component will change due to the acceleration of gravity.

Using basic trigonometry, we can find the horizontal component of the initial velocity, which is given by: u_x = u cos(θ), where θ is the angle of launch. Substituting the given values, we get u_x = u cos(35).

Next, we can use the equation for the vertical component of velocity, which is given by: u_y = u sin(θ) - gt, where g is the acceleration due to gravity. We know that at the maximum height, the vertical component of velocity will be zero. So, we can set u_y = 0 and solve for t.

0 = u sin(35) - gt
t = u sin(35) / g

Now, we can substitute this value of t into the equation for horizontal velocity and solve for u.

45 = u cos(35) * u sin(35) / g
u = √(45g / sin(2θ))

Substituting the given values, we get u = √(45 * 9.8 / sin(70)) = 23.7 m/s

Therefore, the initial velocity required for the football to just clear the goalpost crossbar is approximately 23.7 m/s. We can also check this answer by using the equation for the maximum height of a projectile, which is given by: h = u^2 sin^2(θ)/2g. Substituting the values, we get h = 3.1 m, which is the height of the goalpost crossbar.

In conclusion, to solve this problem, we used the basic principles of kinematics and trigonometry to find the initial velocity required for the football to clear the goal