A student has derived the following nondimensionally homogeneous equation:a=x/t^2 − vt + F/m where v is a velocity's magnitude, a is an acceleration's magnitude, t is a time, m is a mass, F is a force's magnitude, and x is a distance (or length). Which terms are dimensionally homogeneous? Check all that apply. I know the answers are1 2 and 4 but don't know why. explain so I can understand.

Question

A student has derived the following nondimensionally homogeneous equation:a=x/t^2 − vt + F/m where v is a velocity's magnitude, a is an acceleration's magnitude, t is a time, m is a mass, F is a force's magnitude, and x is a distance (or length). Which terms are dimensionally homogeneous? Check all that apply. I know the answers are1 2 and 4 but don't know why. explain so I can understand.

1) x/t^2
2) a
3) F/m
4) vt

Answer 1

1. is meters/second^2

2. is meters / second^2

3. is newtons / kilograms but we all know from Newton's second law that Newtons = kilograms* meters/second^2 so 3. is also meters/second^2

4. (meters / second) * seconds = meters. TILT !

Answer 2

Do your other problem the same way.

Answer 3

answers are1 2 and 3.

Answer 4

To determine which terms are dimensionally homogeneous in the given equation, we need to compare the dimensions of each term.

In physics, there are fundamental dimensions such as length (L), time (T), mass (M), and force (F). Each physical quantity can be expressed as a combination of these dimensions.

Let's analyze each term:

1) x/t^2:
- The variable x represents distance or length, which has the dimension L.
- The variable t represents time, which has the dimension T.
- t^2 has the dimension T^2.
- Therefore, x/t^2 has the dimensions L/(T^2).

2) a:
- The variable a represents acceleration, which has the dimension L/T^2. (Acceleration is the rate of change of velocity with respect to time.)

3) F/m:
- The variable F represents force, which has the dimension F.
- The variable m represents mass, which has the dimension M.
- Therefore, F/m has the dimensions F/M.

4) vt:
- The variable v represents velocity magnitude, which has the dimension L/T.
- The variable t represents time, which has the dimension T.
- Therefore, vt has the dimensions L.

Now, let's match these dimensions with the fundamental dimensions:

1) x/t^2: L/(T^2) - Matches the dimension of acceleration (a).
2) a: L/T^2 - Matches the dimension of acceleration (a).
3) F/m: F/M - Doesn't match the dimension of acceleration (a).
4) vt: L - Doesn't match the dimension of acceleration (a).

From the analysis above, we can conclude that terms 1) x/t^2 and 2) a are dimensionally homogeneous because they have the same dimension, L/T^2. Terms 3) F/m and 4) vt are not dimensionally homogeneous as they have different dimensions from acceleration (a).

Therefore, the correct answers are 1) x/t^2, 2) a, and 4) vt.