Question

A golfer imparts a speed of 30.3 m/s to a ball, and it travels the maximum possible distance before landing on the green. the tee and the green are at the same elevation. (a) how much time does the ball spend in the air? (b) what is the longest hole in one that the golfer can make, if the ball does not roll when it hits the green?

Answer 1

Answer:

a) Time spend by ball in air = 4.368 seconds

b)   Longest hole that golfer can make = 93.59 meter

Explanation:

  Projectile motion has two types of motion Horizontal and Vertical motion.

Vertical motion:

         We have equation of motion, v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration and t is the time taken.

         Considering upward vertical motion of projectile.

         In this case, Initial velocity = vertical component of velocity = u sin θ, acceleration = acceleration due to gravity = -g $m/s^2$ and final velocity = 0 m/s.

        0 = u sin θ - gt

         t = u sin θ/g

    Total time for vertical motion is two times time taken for upward vertical motion of projectile.

    So total travel time of projectile = 2u sin θ/g

Horizontal motion:

  We have equation of motion , $s= ut+\frac{1}{2} at^2$, s is the displacement, u is the initial velocity, a is the acceleration and t is the time.

  In this case Initial velocity = horizontal component of velocity = u cos θ, acceleration = 0 $m/s^2$ and time taken = 2u sin θ /g

 So range of projectile,  $R=ucos\theta*\frac{2u sin\theta}{g} = \frac{u^2sin2\theta}{g}$

a) We have golf ball travels maximum distance, so range is maximum.

                 Maximum range is when, sin 2θ =1

                             =>  θ = 45⁰

    Now we have travel time of projectile, t =  2u sin θ/g  

          Initial velocity = 30.3 m/s and  θ = 45⁰

                     So time spend in air, t = $\frac{2*30.3*sin45}{9.81} =4.368 seconds$

 b) Longest hole that golfer can make = Range of projectile = $\frac{u^2sin2\theta}{g}$

      Longest hole that golfer can make = $\frac{30.3^2sin(2*45)}{9.81}=93.59 meter$