(adsbygoogle = window.adsbygoogle || []).push({}); «Challenging» Projectile Problem

1. The problem statement, all variables and given/known data

The benches of a gallery in a baseball stadium are 1 m wide and 1 m high. A batsman strikes the ball at a level 1 m above the ground and hits a mammoth home run. The ball starts at 35 m/s at an angle of 53 degrees with the horizontal The benches are perpendicular to the plane of motion and the first bench is 110 m from the batsman. On which bench will the ball hit ?

2. Relevant equations

Equations of projectile motion

[tex]R=\frac{u^{2}sin{2\Theta}}{g} [/tex]

[tex]y=xtan\Theta(1-\frac{x}{R})[/tex]

3. The solution i thought of

Ok, I got a solution to this problem but it was very long and tedious (the original method which i thought of) I wonder if there could be some changes to the approach that would give the solution faster. A better method altogether is most welcome :)

Here's my approach

Let h be the height of the ball above the point where it was hit.

First i calculated the range which is [itex]\approx 120 m[/itex]

Then, differentiating eqn of path wrt time, i get-:

[tex]\frac{dh}{dt}=\frac{d(xtan\Theta(1-\frac{x}{R}))}{dt} [/tex]

[tex]\frac{dh}{dx}=tan\Theta(1-\frac{2x}{R})[/tex]

[tex]dh=tan\Theta(1-\frac{x}{60}).dx [/tex]

[tex]\int{dh}=tan\Theta\int{(1-\frac{x}{60}).dx} [/tex]

[tex]h=\frac{4x}{3}(1-\frac{x}{120})[/tex]

Another equation is from the bench thing, so height of bench [itex]y=x-109[/itex]

Equating h and y, i get a quadratic,

[tex]\frac{x^2}{90}-\frac{1}{3x}-109=0[/tex]

The solution to this equation is [itex]\approx 115.3[/itex] , which tells that the bench will be 6th. (and matches the answer)

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# Challenging Projectile Problem

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