A ball is thrown vertically upward with an initial speed of 20 m/s. Two seconds

Question

A ball is thrown vertically upward with an initial speed of 20 m/s. Two seconds later, a stone is thrown vertically (from the same initial height as the ball) with an initial speed of 24 m/s. At what height above the release point will the ball and stone pass each other?

It is 17 m.

S = VoT - (1/2)gT^2

where

S = distance travelled by the body
Vo = initial velocity
T = travel time
g = acceleration due to gravity = 9.8 m/sec^2 (constant)

For the ball,

Sb = 20T - (1/2)(9.8)(T^2)

and for the stone,

Ss = 24(T - 2) + (1/2)(9.8)(T - 2)^2

and for them to pass each other from the release point,

Sb = Ss

and therefore

20T - (1/2)(9.8)T^2 = 24(T - 2) - (1/2)(9.8)(T - 2)^2

20T - 4.9T^2 = 24T - 48 - 4.9(T^2 - 4T + 4)

20T - 4.9T^2 = 24T - 48 - 4.9T^2 + 19.6T - 19.6

Combining like terms,

24T + 19.6T - 20T = 48 +19.6

23.6T = 67.6

T = 2.86 sec.

In other words, the stone will pass the ball 2.86 seconds after the ball has been tossed (or 0.86 sec after the stone has been tossed).

Solving for Sb,

Sb = 20(2.86) - (1/2)(9.8)(2.86^2)

Sb = 17.12 meters --- the ball will be passed by the stone at this elevation above the release point. This is the first option in your given choices.

I am wondering that Why set up (t-2) for Stone?

I think "Two senconds later" means t+2 for the stone.

Explanation / Answer

I assume that your question is about the "t-2".

For the stone the time it is (t-2), because it has a delay respect with the ball. Put it in this way, the stone is thrown TWO SECONDS later, which means that the ball has been moving for two seconds right?. If we use "t" for indicate the time of the ball, it is ok is we say that T=2, because the ball has been moving for that period of time. Now for the stone, if we use "T-2", and change the value of T for 2 (that represents the time of the ball in movement), we get that the stone has benn moving for 0 seconds, that make sense. If we use "T+2", then we will have the value of 4, meaning this that the stone has been moving for 4 seconds, which is wrong.

Hope this is helfull.

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