Question

a ball is dropped from the top of a building . the ball takes 1/4 s to fall past 12 m length of a sign board some distance from the top of the buildings . If the velocity of the ball at the top and at the bottom of thr board are v1 and v2 then
1) v1+v2= 96ms-1 2) v1-v2 =9.8ms-1
3) v1v2 = 1m2s-2 4) v1/v2 =1
please do with full process

Answer 1

Given:

Time taken by ball to fall past the sign board = $\sf \dfrac{1}{4} \ s$

Length of the sign board = 12 m

Velocity of ball at the top of the board = $\sf v_1$

Velocity of ball at the bottom of the board = $\sf v_2$

Answer:

Motion during which ball fall past the sign board:

Initial velocity (u) = $\sf v_1$

Final velocity (v) = $\sf v_2$

Time taken (t) = $\sf \dfrac{1}{4} \ s$

Acceleration due to gravity (g) = 10 m/s²

Height (h) = 12 m

By using $\sf 1^{st}$ equation motion:

$\bf \implies v = u + gt \\ \\ \rm \implies v_2 = v_1 + 10 \times \dfrac{1}{4} \\ \\ \rm \implies v_2 = v_1 + 2.5 \\ \\ \rm \implies v_2 - v_1 = 2.5 \: m {s}^{ - 1} \: \: \: \: ...eq_1$

By using $\sf 3^{rd}$ equation motion:

$\bf \implies {v}^{2} = {u}^{2} + 2gh \\ \\ \rm \implies v_2 ^{2} = v_1 ^{2} + 2 \times 10 \times 12 \\ \\ \rm \implies v_2 ^{2} = v_1 ^{2} + 240 \\ \\ \rm \implies v_2 ^{2} - v_1 ^{2} = 240 \\ \\ \rm \implies (v_2 - v_1 ) (v_2 + v_1 )= 240 \\ \\ \rm Using \ eq_1 : \\ \\ \rm \implies 2.5(v_2 + v_1 )= 240 \\ \\ \rm \implies v_2 + v_1 = \dfrac{240}{2.5} \\ \\ \rm \implies v_2 + v_1 =96 \: m {s}^{ - 1} \: \: \: \: ...eq_2$

Adding $\sf eq_1$ and $\sf eq_2$:

$\rm \implies v_2 - v_1 + v_2 + v_1 =2.5 + 96 \\ \\ \rm \implies 2v_2 = 98.5 \\ \\ \rm \implies v_2 = \dfrac{98.5}{2} \\ \\ \rm \implies v_2 = 49.25 \: m {s}^{ - 1}$

By using $\sf eq_1$:

$\rm \implies 49.25 - v_1 = 2.5 \\ \\ \rm \implies v_1 = 49.25 - 2.5 \\ \\ \rm \implies v_1 = 46.75$

So,

$\rm \implies v_1v_2 = 49.25 \times 46.75 \\ \\ \rm \implies v_1v_2 = 2302.4375 \: m {s}^{ - 1}$

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$\rm \implies \dfrac{v_1}{v_2} = \dfrac{49.25}{46.75} \\ \\ \rm \implies \dfrac{v_1}{v_2} =1.053$

Correct Option: $\boxed{\mathfrak{(1) \ v_1 + v_2 = 96 \ ms^{-1}}}$