Question

kinetic energy of a particle depends on the square of speed of the particle if error in measurement of speed is 40% then the error in measurement of kinetic energy will be​

Answer 1

Kinetic energy depends upon the square of Velocity.

Thus, Relative Error in kinetic energy = Relative Error in mass + 2 x Relative Error in velocity

It should be noted, that 2 is multiplied because the power of the velocity is 2 in the formula of kinetic energy.

Since, about mass, nothing is given.

Thus we will assume the error in mass to be zero.

∴ Relative error in Kinetic energy = 2 × Relative error in speed

∴ ΔK.E./K.E. = 2 × ΔV/V

∴ ΔK.E./K.E. = 2 × 40

∴ ΔK.E./K.E. = 80%

Hence, the Relative error in the measurement of Kinetic energy is 80%.

Hope it helps.

Answer 2

kinetic energy is directly proportional to square of speed of particle.

i.e., $K.E\propto v^2K$

or, K.E = kv² , where k is proportionality constant.

differentiating both sides,

d(K.E) = 2kv.dv

or, d(K.E)/K.E = 2kv.dv/(kv²) [ as K.E = kv²]

or, d(K.E)/K.E = 2dv/v

or, fractional error in K.E =2 × fractional error in v

or, % error in K.E = 2 × % error in v

given, % error in v = 40%

so, % error in K.E = 2 × 40% = 80%

hence, answer should be nearly 80% .

well, above concept is good for less than 10% error.

for better results, you should apply arithmetic calculations . like if we assume K is initial kinetic energy and v is initial velocity then, 40% error means v' = v± 40% of v = 1.4v or 0.6v

then, kf = (1.4/1)² × k = 1.96k and % error of K.E = (1.96k - k)/k × 100 = 96% .

again, kf = (0.6/1)² × k = 0.36k and % error of k.E = (0.36k - k)/k × 100 = -64%

and average of error = (|96| + |-64|)/2 = 80% ,

hence, average error is 80%