Question

A car is travelling along a horizontal road with speed v measured in m/s. The power P measured in watts required to overcome external forces opposing the motion is given by the P = cv + kv^3 where c and k are constants. Use base units to obtain an S.I. unit for constant k

Answer 1

A car is travelling along a horizontal road with speed v measured in m/s. The power P measured in watts required to overcome external forces opposing the motion is given by the P = cv + kv^3 where c and k are constants. Use base units to obtain an S.I. unit for constant k

P = cv + kv³

P = power in watts

1 watt = 1 Joule/Sec

P = Power in Joule/s

Power = (1/2)mv² / t

(1/2)mv² / t  = k v³

=> k = m/vt

=> k = mass /Distance

SI unit of mass = kg

SI unit of Distance = m

=>SI unit of  k = kg/m

(1/2)mv² = cv

=> c = mv

SI unit of c = kgm/s

Answer 2

The power P measured in watts required to overcome external forces opposing the motion is given by the, $P=cv+kv^3$, where c and k are constants.

P denotes power.

we know, power = energy/time

= force × displacement/time

so, unit of power = unit of force × unit of displacement/unit of time,

= Newton. metre/second

we know, 1 Newton = 1 kg.m/s²

so, unit of power, P = kg m/s² × m/s = kg m²/s³

v denotes speed,

so, unit of speed, v = m/s

here, unit of P = unit of (cv)

unit of P = unit of c × unit of v

unit of c = unit of P/unit of v ={ kgm²/s³}/{m/s} = kgm/s²

hence, unit of c = kgm/s² or Newton

again, unit of P = unit of ( kv³)

unit of P = unit of k × unit of v³

unit of k = unit of P/unit of v³

= {kgm²/s³}/{m³/s³}

= kg/m

hence, unit of k = kg/m