Question

The benches of a gallery in a cricket stadium are 1 m wide and 1 m high. A batsman strikes the ball at a level one metre above the ground and hits a mammoth sixer. The ball starts at 35 m/s at an angle of 53° 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?

Answer 1

Given:
Angle of projection of the ball, α = 53°
Width and height of the bench = 1 m
Initial speed of the ball = 35 m/s
Distance of the first bench from the batsman = 110 m
The batsman strikes the ball 1 m above the ground.
Let the ball land on the n^th bench.
∴ y = (n − 1) ...(i)
And,
$x=110+n-1=110+y$



$$\begin{gathered} \mathrm{Now}, \\ y=x\tan \alpha -\left(\frac{g{x}^2{\sec }^2\alpha }{2u^2}\right) \\ \Rightarrow y=\left(110+y\right)\left(\frac{4}{3}\right)-\frac{10\times {\left(110+y\right)}^2\left({\sec }^{2}53°\right)}{2\times {\left(35\right)}^2} \\ =\frac{440}{3}+\frac{4}{3}y-\frac{250{\left(110+y\right)}^2}{18\times {\left(35\right)}^2} \\ =\frac{440}{3}+\frac{4}{3}y-\frac{250{\left(110+y\right)}^2}{18\times {35}^2} \end{gathered}$$
Solving the above equation, we get:
y = 5
⇒ n − 1 = 5
⇒ n = 6
The ball will hit the sixth bench of the gallery.