Given data:

The equation is,

F(x2−x1)=12mV22−12mV12 (1)

Here,

F is the force in N,

m is the mass in kg,

x1, x2 is the distance in m, and

V1, V2 is the velocity in ms,

Formula used:

The SI unit expression in terms of base units as follows,

N=kg⋅ms2

Calculation:

Find the given equation is dimensionally homogenous or not.

Substitute the units N for F, kg for m, m for x1, m for x2, ms for V1 and ms for V2 in equation (1).

N(m−m)=12(kg)(ms)2−12(kg)(ms)2N⋅m=12(kg⋅m2s2−kg⋅m2s2)N⋅m=12kg⋅m2s2 (2)

Here,

12 is constant (unitless).

Equation (2) becomes,

N⋅m=(kg⋅ms2)⋅m (3)

Substitute the unit kg⋅ms2 for N in equation (3).

N⋅m=N⋅m (4)

From equation (4), Left-hand side (LHS) is equal to Right-hand side (RHS). Thus, the given equation is dimensionally homogenous and it is proved.

Conclusion:

Hence, the given equation is dimensionally homogenous and it is proved.

Given data:

The equation is,

F=12mV22−12mV12 (5)

Here,

F is the force in N,

m is the mass in kg, and

V1, V2 is the velocity in ms.

Formula used:

The SI unit expression in terms of base units as follows,

N=kg⋅ms2

Calculation:

Find the given equation is dimensionally homogenous or not.

Substitute the units N for F, kg for m, ms for V1 and ms for V2 in equation (5).

N=12(kg)(ms)2−12(kg)(ms)2N=12(kg⋅m2s2−kg⋅m2s2)N=12kg⋅m2s2 (6)

Here,

12 is constant (unitless).

Equation (6) becomes,

N=(kg⋅ms2)⋅m (7)

Substitute the unit kg⋅ms2 for N in equation (7).

N=N⋅m (8)

From equation (8), Left-hand side (LHS) is not equal to Right-hand side (RHS). Thus, the given equation is not dimensionally homogenous and it is proved.

Conclusion:

Hence, the given equation is not dimensionally homogenous and it is proved.

Given data:

The equation is,

F(V2−V1)=12mx22−12mx12 (9)

Here,

F is the force in N,

m is the mass in kg,

x1, x2 is the distance in m, and

V1, V2 is the velocity in ms,

Formula used:

The SI unit expression in terms of base units as follows,

N=kg⋅ms2

Calculation:

Find the given equation is dimensionally homogenous or not.

Substitute the units N for F, kg for m, m for x1, m for x2, ms for V1 and ms for V2 in equation (9).

N(ms−ms)=12(kg)(m)2−12(kg)(m)2N⋅ms=12(kg⋅m2−kg⋅m2)N⋅ms=12kg⋅m2 (10)

Here,

12 is constant (unitless).

Equation (10) becomes,

N⋅ms=kg⋅m2 (11)

Substitute the unit kg⋅ms2 for N in equation (11).

(kg⋅ms2)⋅ms=kg⋅m2kg⋅m2s3=kg⋅m2 (12)

From equation (12), Left-hand side (LHS) is not equal to Right-hand side (RHS). Thus, the given equation is not dimensionally homogenous and it is proved.

Conclusion:

Hence, the given equation is not dimensionally homogenous and it is proved.

Given data:

The equation is,

F(t2−t1)=mV2−mV1 (13)

Here,

F is the force in N,

m is the mass in kg,

t1, t2 is the time in s, and

V1, V2 is the velocity in ms,

Formula used:

The SI unit expression in terms of base units as follows,

N=kg⋅ms2

Calculation:

Find the given equation is dimensionally homogenous or not.

Substitute the units N for F, kg for m, s for t1, s for t2, ms for V1 and ms for V2 in equation (13).

N(s−s)=(kg)(ms)−(kg)(ms)N⋅s=kg⋅ms−kg⋅msN⋅s=kg⋅ms (14)

Substitute the unit kg⋅ms2 for N in equation (14).

N⋅s=kg⋅ms(kg⋅ms2)⋅s=kg⋅ms

kg⋅ms=kg⋅ms (15)

From equation (15), Left-hand side (LHS) is equal to Right-hand side (RHS). Thus, the given equation is dimensionally homogenous and it is proved.

Conclusion:

Hence, the given equation is dimensionally homogenous and it is proved.