Analysis of an Equation Show that the expression v = at, where v represents speed, a acceleration, and t an instant of time, is dimensionally correct. Dimensions and Units of Four Derived Quantities Quantity Area Volume Speed Acceleration Dimensions L/T L/T2 SI units m2 m3 m/s m/s? U.S. customary units ft2 ft3 ft/s ft/s2 SOLUTION (Use the following as necessary: L and T.) Identify the dimensions of v from the table above: [v] = Identify the dimensions of a from the table above and multiply by the dimensions of t: [at] = Therefore, v = at is dimensionally correct because we have ---Select--- v dimensions on both sides. (If the expression were given as v = at?, it would be dimensionally incorrect. Try it and see!) EXERCISE Suppose you are given the following equation, where x, and x, represent positions at two instants of time, v is a velocity, a, is an acceleration, t is an instant of time, and a, b, and c are integers. X; = x° + v° +ee For what values of a, b, and c is this equation dimensionally correct? Hint a = C =
Analysis of an Equation Show that the expression v = at, where v represents speed, a acceleration, and t an instant of time, is dimensionally correct. Dimensions and Units of Four Derived Quantities Quantity Area Volume Speed Acceleration Dimensions L/T L/T2 SI units m2 m3 m/s m/s? U.S. customary units ft2 ft3 ft/s ft/s2 SOLUTION (Use the following as necessary: L and T.) Identify the dimensions of v from the table above: [v] = Identify the dimensions of a from the table above and multiply by the dimensions of t: [at] = Therefore, v = at is dimensionally correct because we have ---Select--- v dimensions on both sides. (If the expression were given as v = at?, it would be dimensionally incorrect. Try it and see!) EXERCISE Suppose you are given the following equation, where x, and x, represent positions at two instants of time, v is a velocity, a, is an acceleration, t is an instant of time, and a, b, and c are integers. X; = x° + v° +ee For what values of a, b, and c is this equation dimensionally correct? Hint a = C =
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![### Analysis of an Equation
Show that the expression \( v = at \), where \( v \) represents speed, \( a \) acceleration, and \( t \) an instant of time, is dimensionally correct.
#### Dimensions and Units of Four Derived Quantities
| Quantity | Area | Volume | Speed | Acceleration |
|-------------------------|------|--------|---------|--------------|
| Dimensions | \( L^2 \) | \( L^3 \) | \( L/T \) | \( L/T^2 \) |
| SI Units | \( m^2 \) | \( m^3 \) | \( m/s \) | \( m/s^2 \) |
| U.S. Customary Units | \( ft^2 \) | \( ft^3 \) | \( ft/s \) | \( ft/s^2 \) |
### Solution
(Use the following as necessary: \(L\) and \(T\).)
Identify the dimensions of \( v \) from the table above:
\[ [v] = \boxed{L/T} \]
Identify the dimensions of \( a \) from the table above and multiply by the dimensions of \( t \):
\[ [at] = \left( \boxed{L/T^2} \right) T = \boxed{L/T} \]
Therefore, \( v = at \) is dimensionally correct because we have \(\boxed{\text{Select}}\) dimensions on both sides. (If the expression were given as \( v = at^2 \), it would be dimensionally incorrect. Try it and see!)
### Exercise
Suppose you are given the following equation, where \( x_f \) and \( x_i \) represent positions at two instants of time, \( v_{xi} \) is a velocity, \( a_x \) is an acceleration, \( t \) is an instant of time, and \( a \), \( b \), and \( c \) are integers.
\[ x_f = x_f^a + v_{xi} x_f^b + \frac{1}{2} a_x t^c \]
For what values of \( a \), \( b \), and \( c \) is this equation dimensionally correct?
\[ a = \boxed{} \]
\[ b = \boxed{} \]
\[ c =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe5add6b-1715-412c-ae9e-e5ef4329e8a2%2Fa67c0968-e426-4ecf-a4e0-028963a84bec%2Fadtw0s_processed.png&w=3840&q=75)
Transcribed Image Text:### Analysis of an Equation
Show that the expression \( v = at \), where \( v \) represents speed, \( a \) acceleration, and \( t \) an instant of time, is dimensionally correct.
#### Dimensions and Units of Four Derived Quantities
| Quantity | Area | Volume | Speed | Acceleration |
|-------------------------|------|--------|---------|--------------|
| Dimensions | \( L^2 \) | \( L^3 \) | \( L/T \) | \( L/T^2 \) |
| SI Units | \( m^2 \) | \( m^3 \) | \( m/s \) | \( m/s^2 \) |
| U.S. Customary Units | \( ft^2 \) | \( ft^3 \) | \( ft/s \) | \( ft/s^2 \) |
### Solution
(Use the following as necessary: \(L\) and \(T\).)
Identify the dimensions of \( v \) from the table above:
\[ [v] = \boxed{L/T} \]
Identify the dimensions of \( a \) from the table above and multiply by the dimensions of \( t \):
\[ [at] = \left( \boxed{L/T^2} \right) T = \boxed{L/T} \]
Therefore, \( v = at \) is dimensionally correct because we have \(\boxed{\text{Select}}\) dimensions on both sides. (If the expression were given as \( v = at^2 \), it would be dimensionally incorrect. Try it and see!)
### Exercise
Suppose you are given the following equation, where \( x_f \) and \( x_i \) represent positions at two instants of time, \( v_{xi} \) is a velocity, \( a_x \) is an acceleration, \( t \) is an instant of time, and \( a \), \( b \), and \( c \) are integers.
\[ x_f = x_f^a + v_{xi} x_f^b + \frac{1}{2} a_x t^c \]
For what values of \( a \), \( b \), and \( c \) is this equation dimensionally correct?
\[ a = \boxed{} \]
\[ b = \boxed{} \]
\[ c =
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