Question

A body dropped from a height h with initial velocity zero, strikes the ground with velocity $3m/s$. Another body of the same mass is dropped from the height h with an initial velocity of $4m/s.$Find the velocity with which it strikes the ground.
A. $3m/s~$
B. $4m/s~~$
C. $~5m/s~$
D. $12m/s~$

Answer 1

While dealing with problems based on one dimensional Motion of an object in a straight line, we usually come across three kinematic variables namely velocity$\left( v \right)$ , position$\left( s \right)$ and time$\left( t \right)$ and to find one when others are already provided , we use the three equations of motion that are basically equations that describe the behavior of a physical system in terms of the given variables and each could be characterized by an entanglement of two.

Formula used:
The three equations of motion are:
$v=u+gt$
$h=ut+\dfrac{1}{2}g{{t}^{2}}$
${{v}^{2}}-{{u}^{2}}=2gh$
Where $u=Initial\text{ }velocity$
$v=Final\text{ }velocity$
$g=acceleration\text{ }due\text{ }to\text{ }gravity$
$h=height$
$t=time$

Complete step-by-step answer:
For the first body:
Since it’s given in the question that the first body is dropped,
Therefore;
Initial velocity of the body: $0m/s$
From the question:
Final velocity of the body: $3m/s$
Therefore;
$v=u+gt$ where $g=10m/{{s}^{2}}$
After substituting the values:
$3=10t$
$\therefore t=0.3s$
Hence the time taken by the first body to obtain the final velocity of $3m/s$is $0.3s$

The distance covered by the body: $'h'$
Therefore;
$h=ut+\dfrac{1}{2}g{{t}^{2}}$ where $g=10m/{{s}^{2}}$
After substituting the values:
$h=0+\dfrac{1}{2}(10){{(0.3)}^{2}}$
$h=0.45m$
For the second body:
It’s given from the question that:
Initial velocity of the body:$4m/s$
The distance it will be covering is $0.45m$
Therefore,
From the third equation of motion, we can obtain the final velocity of the second body. i.e.
${{v}^{2}}-{{u}^{2}}=2gh$
After rearranging the terms and substituting the known values, we get:
${{v}^{2}}-{{(4)}^{2}}=2(10)(0.45)$
${{v}^{2}}=25$
$\Rightarrow v=5m/s$
Therefore, the final velocity of the second body is $5m/s$ and so the correct option will be option C.

So, the correct answer is “Option C”.

Note: It should be noted that the acceleration due to gravity $'g'$ is taken as a positive term here as the direction of motion of both the bodies are in the direction of the gravitational acceleration.
Special care should be taken in calculations involving exponential powers as Students might accidentally make small errors in the exponential powers while calculating and end up getting incorrect results.
Do not forget to mention the units after each equation because it might lead to deduction of marks and also errors regarding conversions in subsequent steps.