To tackle your questions, let's break them down one at a time, starting with the dimensional formula for EC in the speed equation. The equation you provided is:
Understanding the Speed Equation
The speed of an aircraft is given by:
v = A + ABt + ABClog(Dx) + ED
In this equation, we need to identify the dimensions of each term to find the dimensional formula for EC. Here, v represents speed, which has the dimensional formula of [L][T-1].
Breaking Down the Terms
- A: This term must also have the dimensions of speed, so it has the same dimensional formula as v, which is [L][T-1].
- ABt: Here, A has dimensions of speed, and B must have dimensions of [L][T-2] (acceleration) to ensure that the product ABt has dimensions of speed. Thus, t has dimensions of [T]. Therefore, ABt has dimensions of [L][T-1].
- ABClog(Dx): The term log(Dx) is dimensionless, so ABClog(Dx) must also have dimensions of speed. Hence, AB must have dimensions of [L][T-1].
- ED: For this term to also represent speed, E must have dimensions of [L][T-1] when multiplied by D, which must be dimensionless.
Finding the Dimensional Formula for EC
Since we established that all terms must equate to the dimensions of speed, we can conclude that:
EC must also have dimensions of [L][T-1].
Analyzing the Error in Kinetic Energy
Now, let's move on to your second question regarding the error in the measurement of kinetic energy when the percentage error in momentum is given as +100% and the mass is error-free.
Understanding Momentum and Kinetic Energy
The formulas for momentum (p) and kinetic energy (KE) are:
- Momentum: p = mv
- Kinetic Energy: KE = (1/2)mv2
Calculating the Error in Kinetic Energy
Given that the percentage error in momentum is +100%, this means that the measured momentum is twice the actual momentum. If we denote the actual momentum as pactual and the measured momentum as pmeasured, we have:
pmeasured = 2 * pactual
Now, since kinetic energy depends on the square of the velocity (which is derived from momentum), we can express the kinetic energy in terms of momentum:
KE = (p2)/(2m)
Impact of Momentum Error on Kinetic Energy
When the momentum is doubled, the kinetic energy becomes:
KEmeasured = (2pactual)2/(2m) = (4pactual2)/(2m) = 2 * (pactual2)/(2m) = 2 * KEactual
This indicates that the measured kinetic energy is twice the actual kinetic energy. Therefore, the percentage error in kinetic energy is:
Percentage Error in KE = (KEmeasured - KEactual) / KEactual * 100% = (2KEactual - KEactual) / KEactual * 100% = 100%
Final Thoughts
In summary, the dimensional formula for EC is [L][T-1], and the error in the measurement of kinetic energy, given a +100% error in momentum, is also +100%. This illustrates how errors in fundamental quantities can significantly impact derived quantities like kinetic energy.
