Question

Bruce Springsteen throws his guitar off stage to a tech that is waiting to catch it. Approximate the guitar as a point particle. You may ignore the heights of Bruce and his tech. The Boss lets the guitar fly from the edge of the stage at an angle of β off the vertical with an initial velocity of V0 . The tech is horizontally separated from Bruce by a distance of 2F and the guitar is caught F below its launch point. A.) (3pts) What is the maximum height above the ground that the guitar reaches? Your answer should be in terms of V0 , F, g and β only. B.) (3pts) How much time does the guitar spend in the air? Your answer should be in terms of V0 , F, g and β only. C.)(2pts) What are the x and y components of the velocity of the guitar just before the tech catches it? Answers should be in terms of V0 , g, β, and F only. D.)(2pts) If β increases, will the time of flight increase, decrease, or stay the same?

Answer 1

let,

initial velocity of an object at time

final velocity of that object at time ( directed parallel to the direction of velocity )

displacement of that object along the line parallel to the direction of velocity ( or )

acceleration of that object directed along the line parallel to the direction of ( or )

In this problem we need to use the fallowing kinematic relations ( relations between above mentioned )

_____________________________________ relation 1

  _____________________________________ relation 2

_____________________________________ relation 3

we have,

magnitude of velocity with which the guitar is thrown

angle with respect to vertical with which the guitar is thrown

In above figure,

Vertical component of velocity

Horizontal component of velocity

Note,

Since there is only gravitational force acting downwards on the guitar along the vertical, and no force acting on the guitar along the horizontal when it is in the air,

Only the vertical component of the velocity of guitar will change along the path and horizontal component of the velocity will not change,

A)

Let,

the maximum height  guitar reaches in the air

Using,

( -ve sign indicates that the acceleration is in the opposite direction of velocity )

( when guitar reaches maximum height its vertical velocity becomes zero )

In relation 2 we get,

So we get, maximum height as,

B)

we have,

total horizontal distance traveled by guitar distance between edge of the stage and tech​

Horizontal component of velocity

We will use the fact that as mentioned earlier the horizontal component of the velocity stays constant.

Let,

Time for which the guitar stays in the air   Total time of flight of guitar

Since, Velocity is given as distance divided by time required to travel that distance we get,

So, we get the maximum time for which guitar stays in the air ( time of flight ) as,

___________________________________________________ relation 4

C)

as mentioned earlier the horizontal component of the velocity stays constant throughout the trajectory.

so, the horizontal component of the velocity just before tech catches the guitar

using,

in relation 3 we get the vertical component of velocity   just before the tech catches it as,

D)

From part B we have the relation for time of flight ( relation 4 ) as,

as   increases   increases, and thereby denominator of above equation increases,

and thereby as denominator increases the time of flight of the guitar decreases

So, In conclusion,

as increases the time of flight decreases.