Question

A ball of mass 4kg and another ball of mass 8kg is dropped together from a 60 feet tall building. After a fall of 30 feet each towards the earth, their respective kinetic energies will be in the ratio of?
A.$\sqrt{2}:1$
B.$1:4$
C.$1:2$
D.$1:\sqrt{2}$

Answer 1

Study about Newton's laws of motion and where we can apply them. Learn about the kinetic energy of a system and about the conservation of energy theorems.

Complete answer:
The kinetic energy of an object of mass m and velocity v is given by the equation,
$K.E.=\dfrac{1}{2}m{{v}^{2}}$
In the question, it is given that the mass of the balls are 4kg and 8kg and both the balls are dropped from the same height of 60 feet. We need to calculate the ratio of kinetic energies of the two balls.
To find the kinetic energies we need to find the velocities of the two balls at the height of 30 feet after dropping. For this we will use the Newton’s law of motion, where,
${{v}^{2}}={{u}^{2}}+2as$
Where, v is the final velocity, u is the initial velocity, a is the acceleration of the object and s is the distance travelled.
Since, both the masses are free falling i.e. there is no external force acting on the masses, the final velocity will be independent of the mass of the object. That is at the fall of 30 feet they will have the same velocity v.
So, the kinetic energy of the ball of mass 4kg is,
$\begin{aligned} & K.E{{.}_{1}}=\dfrac{1}{2}\times 4\times {{v}^{2}} \\\ & K.E{{.}_{1}}={2{v}^{2}} \\\ \end{aligned}$
The kinetic energy of the ball of mass 8kg is,
$\begin{aligned} & K.E{{.}_{2}}=\dfrac{1}{2}\times 8\times {{v}^{2}} \\\ & K.E{{.}_{2}}=4{{v}^{2}} \\\ \end{aligned}$
The ratio of the kinetic energy is,
$\begin{aligned} & K.E{{.}_{1}}:K.E{{.}_{1}}=\dfrac{2{{v}^{2}}}{4{{v}^{2}}} \\\ & K.E{{.}_{1}}:K.E{{.}_{1}}=\dfrac{1}{2}=1:2 \\\ \end{aligned}$

So, the correct option is (C).

Note:
If an object is not under any external influence or force, the object will free fall. The free falling bodies will fall at the same rate irrespective of their mass since the gravitational field of earth causes all objects to fall under the same acceleration of $9.8m{{s}^{-2}}$.