Question

An automobile of constant power p travels a distance x in time t kinetic energy of the automobile is directly proportional to

Answer 1

An automobile of constant power P travels a distance x in time t.

To find : The kinetic energy of the automobile is directly proportional to ......

solution : it has given that power = constant.

i.e., P = F.v = constant

where F is force i.e., F = m dv/dt

and v is speed.

so, m dv/dt v = k

⇒m ∫vdv = k ∫dt

⇒mv²/2 = kt

⇒v² = 2kt/m

⇒v = √{2kt/m}

as v is the rate of change of distance with respect to time.

i.e., v = dx/dt = √{2k/m} t½

⇒∫dx = √{2k/m} ∫t½ dt

⇒x = (2/3)√{2kt/m} t³/²

i.e., x ∝ t³/²

kinetic energy of automatic = power × time

= p × t

here p is constant. so, kinetic energy is directly proportional to time.

Therefore kinetic energy of the automobile is directly proportional to time.