Question

A ball is thrown vertically upwards with an intial Velocity such that it can reach a maximum height of 15 m. If at the same instance, a stone is dropped from height of 15 m,find the Ratio of distance travelled by them when they cross each other.​

Answer 1

Let the two meet at time t seconds.

According to attachment :-

Let distance travelled by stone be x m and distance travelled by ball will be (15-x) when they meet.

$\rule{200}{2}$

For Stone :-

Initial velocity = 0m/sec

Acceleration = (10)m/sec²

Time =t seconds

Distance = x m

Hence,

$s = ut + \frac{1}{2}a {t}^{2} \\ \\ \\x = \frac{1}{2} \times 10 \times {t}^{2} \\ \\ \\ x = 5 {t}^{2}$

$\rule{200}{2}$

For the ball :-

Maximum height = 15 m

Initial velocity = u

Final velocity = 0 m/sec

Acceleration =(-10)m/sec²

Now,

$2as = {v}^{2} - {u}^{2} \\ \\ \\2( - 10)15 = {0}^{2} - {u}^{2} \\ \\ \\ {u}^{2} = 300 \\ \\ \\u = 10 \sqrt{3}$

$\rule{100}{2}$

Now,

For ball when it crosses the stone:-

Distance = 15-x

Initial velocity = 10root3 m/sec

Time = t seconds

Acceleration =(-10)m/sec²

Hence

$s = ut + \frac{1}{2}g {t}^{2} \\ \\ \\15 - x = 10 \sqrt{3}t - 5 {t}^{2} \\ \\ \\x = 5 {t}^{2} - 10 \sqrt{3}t + 15$

$\rule{200}{2}$

From the two equations we get :-

$5 {t}^{2} - 10 \sqrt{3}t + 15 = 5 {t}^{2} \\ \\ \\10 \sqrt{3}t = 15 \\ \\ \\t = \frac{15}{10 \sqrt{3} } \\ \\ \\t = \frac{ \sqrt{3} }{2}$

Hence,

They meet each other at t= root3/2 seconds.

$\rule{200}{2}$

Distance travelled by stone will be :-

$x = 5 {t}^{2} \\ \\ \\x = 5 \times \frac{3}{4} \\ \\x = \frac{15}{4} \\ \\ \\x = 3.75 \: \: m$

Distance covered by ball will be

$= 15 - x \\ \\ = 15 - 3.75 \\ \\ = 11.25 \: \: m$

$\rule{200}{2}$

Ratio of their distances covered will be :-

$\frac{ball}{stone} = \frac{11.25}{3.75} \\ \\ \\ = \frac{1125}{375} \\ \\ \\ = \frac{3}{1} \\ \\ \\ = 3:1$

Hence,

Ratio will be 3:1

$\rule{200}{2}$

Answer 2

Answer :-

Ratio of distance travelled by them when they cross each other is 3 : 1.

Solution :-

Refer to the Attachment.

Hope it Helps !!!!