Question

The energy required to accelerate a car from 10m/s to 20m/s is how many times the energy required to accelerate the car from the rest to 10m/s:
A. Equal
B. 4 times
C. 2 times
D. 3 times

Answer 1

Hint: Energy in physics is defined as the ability or capacity of any object to do some work. Energy can be there in many forms in a system such as kinetic energy, heat energy, potential energy, thermal energy etc.

Formula Used:
Since it is given that energy is required to accelerate the car, that means kinetic energy is used here. Kinetic energy is the energy possessed by any object due to it being in motion. Mathematically, the formula for kinetic energy is given by,
\[E = \dfrac{1}{2}m{v^2}\]
Where ‘m’ is the mass of the object and ‘v’ is the velocity of the object.

Complete step by step solution:
Given that initially the car was moving at 10m/s and energy was required to increase it to 20 m/s. That means there is change in kinetic energy as the velocity increases. The change in kinetic energy will be written as,
\[{E_1} = \dfrac{1}{2}m(v_2^2 - v_1^2)\]
\[\Rightarrow {E_1} = \dfrac{1}{2} \times m[{(20)^2} - {(10)^2}]\]
\[\Rightarrow {E_1} = \dfrac{1}{2}m(400 - 100)\]
\[\Rightarrow {E_1} = \dfrac{1}{2}(300)m\]

Similarly, there is change in kinetic energy and energy required to accelerate the car from rest to 10 m/s is given by
\[{E_2} = \dfrac{1}{2}m(v_2^2 - v_1^2)\]
\[\Rightarrow {E_2} = \dfrac{1}{2}m[{(10)^2} - {(0)^2}]\]
\[\Rightarrow {E_2} = \dfrac{1}{2}(100)m\]

Taking their ratios and solving we get
\[\dfrac{{{E_1}}}{{{E_2}}} = \dfrac{{\dfrac{1}{2} \times 300m}}{{\dfrac{1}{2} \times 100m}}\]
\[\therefore \dfrac{{{E_1}}}{{{E_2}}} = 3\]
The energy required to accelerate the object from rest to 10 m/s is 3 times the energy required to accelerate the car from 10 m/s to 20 m/s.

Hence, option D is the correct answer.

Note: Whenever work is done on any object, some force is applied on it. As a result there is some change and if the object changes its speed, then that change is transferred in the form of kinetic energy. Kinetic energy of an object depends on mass and velocity and has only magnitude but no direction.