Answer
Verified
91.5k+ views
Hint Directly use the formula of the mechanical power to find the equation for velocity. Mechanical power is defined as the rate of doing work given as the ratio of work done to time. Using this formula, find the suitable expression for velocity.
Complete Step By Step Solution
Mechanical power is defined as the amount of work done per unit time. We know that work done on an object is given as the product of force applied on the object and the displacement experienced by the object when the force is applied. Mathematically given as,
\[W = F \times s\]
Now, mechanical power is given as ratio of work done by time, which can also be written as,
\[P = \dfrac{W}{t} = \dfrac{{F \times s}}{t} = Fv\] , Since displacement per time is equal to velocity of the object. Using this formula in our question, it is given that the car moves \[x\] distance in speed \[v\]. Now , power can be calculated as ,
\[ \Rightarrow P = Fv\]
We know that force experienced by an object can be written as the product of its mass and acceleration. Using this in above equation we get,
\[ \Rightarrow P = mav\]
We know that acceleration of a body can be defined as the rate of change of velocity per unit time. Using this in the above equation we get,
\[ \Rightarrow P = m\dfrac{{dv}}{{dt}} \times v\]
\[ \Rightarrow \dfrac{P}{m} = v\dfrac{{dv}}{{dt}}\]
Now, splitting the \[\dfrac{{dv}}{{dt}}\]term as a product of \[dx\], we obtain
\[ \Rightarrow \dfrac{P}{m} = v\dfrac{{dv}}{{dx}} \times \dfrac{{dx}}{{dt}}\]
Now we know that the rate of change of displacement with respect to time is velocity. Using this, we get
\[ \Rightarrow \dfrac{P}{m} = {v^2}\dfrac{{dv}}{{dx}}\]
Taking the \[dx\] to the other side and integrating on both sides we get,
\[ \Rightarrow \int {\dfrac{P}{m}dx} = \int {{v^2}dv} \]
Post integrating, we obtain
\[ \Rightarrow \dfrac{P}{m}x = \dfrac{{{v^3}}}{3}\]
\[ \Rightarrow {v^3} = \dfrac{{3Px}}{m}\]
Applying cube root on both sides, we get
\[ \Rightarrow v = {(\dfrac{{3xP}}{m})^{1/3}}\]
Thus, option (c) is the right answer for the given question.
Note In general kinematics, a work is said to be done on the object when there is a force acting upon a body causing it to get displaced from its original point. According to the work energy theorem, the work done by all forces acting on a body is equal to the change in kinetic energy of the body.
Complete Step By Step Solution
Mechanical power is defined as the amount of work done per unit time. We know that work done on an object is given as the product of force applied on the object and the displacement experienced by the object when the force is applied. Mathematically given as,
\[W = F \times s\]
Now, mechanical power is given as ratio of work done by time, which can also be written as,
\[P = \dfrac{W}{t} = \dfrac{{F \times s}}{t} = Fv\] , Since displacement per time is equal to velocity of the object. Using this formula in our question, it is given that the car moves \[x\] distance in speed \[v\]. Now , power can be calculated as ,
\[ \Rightarrow P = Fv\]
We know that force experienced by an object can be written as the product of its mass and acceleration. Using this in above equation we get,
\[ \Rightarrow P = mav\]
We know that acceleration of a body can be defined as the rate of change of velocity per unit time. Using this in the above equation we get,
\[ \Rightarrow P = m\dfrac{{dv}}{{dt}} \times v\]
\[ \Rightarrow \dfrac{P}{m} = v\dfrac{{dv}}{{dt}}\]
Now, splitting the \[\dfrac{{dv}}{{dt}}\]term as a product of \[dx\], we obtain
\[ \Rightarrow \dfrac{P}{m} = v\dfrac{{dv}}{{dx}} \times \dfrac{{dx}}{{dt}}\]
Now we know that the rate of change of displacement with respect to time is velocity. Using this, we get
\[ \Rightarrow \dfrac{P}{m} = {v^2}\dfrac{{dv}}{{dx}}\]
Taking the \[dx\] to the other side and integrating on both sides we get,
\[ \Rightarrow \int {\dfrac{P}{m}dx} = \int {{v^2}dv} \]
Post integrating, we obtain
\[ \Rightarrow \dfrac{P}{m}x = \dfrac{{{v^3}}}{3}\]
\[ \Rightarrow {v^3} = \dfrac{{3Px}}{m}\]
Applying cube root on both sides, we get
\[ \Rightarrow v = {(\dfrac{{3xP}}{m})^{1/3}}\]
Thus, option (c) is the right answer for the given question.
Note In general kinematics, a work is said to be done on the object when there is a force acting upon a body causing it to get displaced from its original point. According to the work energy theorem, the work done by all forces acting on a body is equal to the change in kinetic energy of the body.
Recently Updated Pages
Name the scale on which the destructive energy of an class 11 physics JEE_Main
Write an article on the need and importance of sports class 10 english JEE_Main
Choose the exact meaning of the given idiomphrase The class 9 english JEE_Main
Choose the one which best expresses the meaning of class 9 english JEE_Main
What does a hydrometer consist of A A cylindrical stem class 9 physics JEE_Main
A motorcyclist of mass m is to negotiate a curve of class 9 physics JEE_Main
Other Pages
The vapour pressure of pure A is 10 torr and at the class 12 chemistry JEE_Main
Electric field due to uniformly charged sphere class 12 physics JEE_Main
3 mole of gas X and 2 moles of gas Y enters from the class 11 physics JEE_Main
If a wire of resistance R is stretched to double of class 12 physics JEE_Main
Derive an expression for maximum speed of a car on class 11 physics JEE_Main
Velocity of car at t 0 is u moves with a constant acceleration class 11 physics JEE_Main