An automobile of mass m accelerates starting | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Power $P$ is given by $P = Fv = (ma)v = m \left(\frac{dv}{dt}\right)v$.Since $P$ is constant,$P = mv \frac{dv}{dt}$.Integrating with respect to ti

Vedclass · Accepted Answer

$\left(\frac{8P}{9m}\right)^{1/2} t^{3/2}$

Vedclass · Answer

(A) Power $P$ is given by $P = Fv = (ma)v = m \left(\frac{dv}{dt}\right)v$.Since $P$ is constant,$P = mv \frac{dv}{dt}$.Integrating with respect to time: $\int_{0}^{v} mv \, dv = \int_{0}^{t} P \, dt$.$\frac{1}{2}mv^2 = Pt \implies v = \sqrt{\frac{2Pt}{m}} = \left(\frac{2P}{m}\right)^{1/2} t^{1/2}$.Since $v = \frac{dx}{dt}$,we have $\frac{dx}{dt} = \left(\frac{2P}{m}\right)^{1/2} t^{1/2}$.Integrating with respect to position and time: $\int_{0}^{x} dx = \int_{0}^{t} \left(\frac{2P}{m}\right)^{1/2} t^{1/2} dt$.$x = \left(\frac{2P}{m}\right)^{1/2} \left[ \frac{t^{3/2}}{3/2} \right] = \left(\frac{2P}{m}\right)^{1/2} \cdot \frac{2}{3} t^{3/2}$.$x = \left( \frac{4}{9} \cdot \frac{2P}{m} \right)^{1/2} t^{3/2} = \left( \frac{8P}{9m} \right)^{1/2} t^{3/2}$.

An automobile of mass $m$ accelerates starting from the origin and is initially at rest,while the engine supplies constant power $P$. The position $x$ as a function of time $t$ is given by:

Similar Questions

$A$ particle of mass $m$ is driven by a machine that delivers a constant power $k$ watts. If the particle starts from rest,the force on the particle at time $t$ is:

$A$ body moves along a straight line due to a constant power delivered by a machine. The distance covered by the body in time $t$ is proportional to:

$A$ motor pump lifts $6$ tonnes of water from a well of depth $25 \ m$ to the first floor of height $35 \ m$ from the ground floor in $20$ minutes. The power of the pump (in $kW$) is: $[g=10 \ ms^{-2}]$

Water falls on the blades of a turbine at a rate of $100 \ kg/s$ from a height of $100 \ m$. The power generated is . . . . . . $kW$.

$A$ car of mass $m$ has an engine which can deliver power $P$. The minimum time in which the car can be accelerated from rest to a speed $v$ is .............

Vedclass Test Series

Exam Paper Generator

Online Exam Module