Which of the equations below are dimensionally correct? (where v is velocity, x is

Question

position, a is acceleration, and t is time)𝑥
=
𝑥
𝑖
+
𝑣
𝑥
𝑖
𝑡
+
1
2
𝑎
𝑡
2
x=x
i
​
+v
xi
​
t+
2
1
​
at
2

Answer 1

To determine if the equation

$$x=x_i+v_{xi}t+\frac{1}{2}a{t}^2$$

is dimensionally correct, we need to analyze the dimensions of each term in the equation.

Let’s define the dimensions of the variables involved:

• $x$ (position) has the dimension of length: $[L]$
• $x_i$ (initial position) also has the dimension of length: $[L]$
• $v_{xi}$ (initial velocity) has the dimension of length per time: $[L][T^{-1}]$
• $t$ (time) has the dimension of time: $[T]$
• $a$ (acceleration) has the dimension of length per time squared: $[L][T^{-2}]$

Now, we evaluate the dimensions of each term:

1. The left side $x$ has the dimension: $$[L]$$

2. The term $x_i$ also has the dimension: $$[L]$$

3. The term $v_{xi}t$ has dimensions: $$[L][T^{-1}]\cdot [T]=[L]$$

4. The term $\frac{1}{2}a{t}^2$ has dimensions: $$\frac{1}{2}\cdot [L][T^{-2}]\cdot [T^2]=[L]$$

Now, putting it all together:

The right side of the equation is: $$x_i+v_{xi}t+\frac{1}{2}a{t}^2$$ This can be expressed as: $$[L]+[L]+[L]=[L]$$

Since both sides of the equation have the dimension of length $[L]$, we can conclude that the equation is dimensionally correct.

Final conclusion: The equation $x=x_i+v_{xi}t+\frac{1}{2}a{t}^2$ is dimensionally correct.