Question

A ball is dropped from a building of height 45 m. Simultaneously another identical ball B is thrown up with a speed \[50\,m{{s}^{-1}}\]. The relative speed of ball B w.r.t ball A at any instant of time is (Take \[g=10\,m{{s}^{-2}}\]).
\[\begin{align}
  & A.\,0 \\
 & B.\,10\,m{{s}^{-1}} \\
 & C.\,25\,m{{s}^{-1}} \\
 & D.\,50\,m{{s}^{-1}} \\
\end{align}\]

Answer 1

Hint: The velocity of an object relative to the other object that might be stationary or moving with either same or a different velocity. The magnitude of the velocity of objects with respect to one another will be more than the individual velocities.
Formula used:
\[v=u-gt\]
\[{{v}_{12}}={{v}_{1}}-{{v}_{2}}\]
\[{{v}_{21}}={{v}_{2}}-{{v}_{1}}\]

Complete step by step answer:
The relative velocity can be defined as the rate of change of relative position of one object concerning another.
The relative velocity of object 1 concerning to object 2 is mathematically represented as follows. \[{{v}_{12}}={{v}_{1}}-{{v}_{2}}\]. Similarly, the relative velocity of object 2 concerning to object 1 is mathematically represented as follows. \[{{v}_{21}}={{v}_{2}}-{{v}_{1}}\]. Irrespective of the direction, the magnitude of these relative velocities will be equal, that is, \[\left| {{v}_{12}} \right|=\left| {{v}_{21}} \right|\].
From the given information we have the data as follows.
A ball is dropped from a building of height 45 m. Simultaneously another identical ball B is thrown up with a speed \[50\,m{{s}^{-1}}\].
The formula that we will be using to find the relative velocity is,
\[v=u-gt\] 
Where v is the final velocity, u is the initial velocity, g is the acceleration due to gravity and t is the time taken.
Consider the ball A.
\[\begin{align}
  & {{v}_{A}}=u-gt \\
 & \Rightarrow {{v}_{A}}=0-10\times t \\
 & \therefore {{v}_{A}}=-10t \\
\end{align}\]
Consider the ball B,
\[\begin{align}
  & {{v}_{B}}=u-gt \\
 & \Rightarrow {{v}_{B}}=50-10\times t \\
 & \therefore {{v}_{B}}=50-10t \\
\end{align}\]
The relative speed of ball B w.r.t ball A is,
\[\begin{align}
  & {{v}_{BA}}={{v}_{B}}-{{v}_{A}} \\
 & \Rightarrow {{v}_{BA}}=(50-10t)-(-10t) \\
 & \Rightarrow {{v}_{BA}}=50-10t+10t \\
 & \therefore {{v}_{BA}}=50{m}/{s}\; \\
\end{align}\]

So, the correct answer is “Option D”.

Note: The motion will be relative as it has to be observed concerning an observer. The measurement of the velocity of an object concerning the other observer. Considering the motion equation, the acceleration is replaced with the acceleration due to gravity, as the acceleration due to gravity acts on the object when their motion is vertical.