Question

The benches of a gallery in a cricket stadium are 1m wide & 1m high. A batsman strikes the ball at a level one mete above the ground & hits a mammoth sixer . The ball starts at35m/s at an angle of 53 degree with thehorizontal. T he benches are perpendicular to the plain of motion & the first bench is 110m from the batsman. On which bench the ball hit.

Answer 1

Answer on Question 3655, Physics, Mechanics | Kinematics | Dynamics

$V_{H}$ - horizontal velocity

$V_{V}$ - vertical velocity

$V_{0} = 35 \mathrm{~m} / \mathrm{s}, \alpha = 53{}^{\circ}, H_{0} = 1 \mathrm{~m}, L = 110 \mathrm{~m}$

$V_{H} = V_{0}\cos (\alpha) = 35\mathrm{m / s}^{*}\cos (53{}^{\circ}) = 35\mathrm{m / s}^{*}0.6018 = 21.0630\mathrm{m / s}$

$V_{V}(t) = V_{0}\sin (\alpha) - gt = 35^{*}\sin (53{}^{\circ}) - 9.8t = 35^{*}0.7986 - 9.8t = (27.9510 - 9.8t)m / s$

Let's find $H_{\max}$ .

$V_{V}(T) = 0$

$V_{0}\sin (\alpha) = gT => T = (V_{0}\sin (\alpha)) / g = (35\mathrm{m / s}^{*}0.7986) / (9.8\mathrm{m / s}^{2}) = 2.8521$ (s)

$H_{\max} = H_0 + TV_V(0) - (gT^2) / 2 = 1 + 2.8521*27.9510 - (9.8*2.8521^2) / 2 = 40.8601(m)$

Let $t_f$ be the time when ball moved 110m horizontal.

$t_f = L / V_H = 110 / 21.0630 = 5.2224(s)$

$H_{f} = H_{\max} - (g(t_{f} - T)^{2}) / 2 = 40.8601 - (9.8*(5.2224 - 2.8521)^{2}) / 2 = 13.3303m$

So, the ball will fly 12.3303 meters higher the bench.

Now let write some relations which will help us to define on which bench ball will hit:

$$h = H _ {m a x} - \frac {g t ^ {2}}{2},$$ $$L _ {0} + x = V _ {H} t,$$

where $L_0 = L - V_H T$ is a horizontal distance from the highest point of trajectory to the first bench. According to this definition the point $h = x$ will show us in what bench the ball fell:

$$\frac {g t ^ {2}}{2} + V _ {H} t - (L _ {0} + H _ {m a x}) = 0 \rightarrow T _ {b e n c h} = 2. 6 5 s$$

Then total time of flight before the conditions come true is $t_{tot} = T + T_{bench} = 5.5 \, \text{s}$ . And total distance accounted from the starting point is $L_{tot} = V_H T_{tot} = 21 \cdot 5.5 = 115.6 \, \text{m}$ . It is clearly shows that the ball will hit sixth bench.

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