can someone explain the answers to the question:
Consider the following set of equations, where s, s0 and r have units of length, t has units of time, v has units of velocity, g and a have units of acceleration. Take k to be dimensionless.Identify if these are dimensionally correct or incorrect.
1)v^2= 2as + kav
2)s = s0 + kat^2
3)a =v^2/r+kv/t
4)t =v/a
5)t = k square root of s/g
1)v^2= 2as + kav
m²/s²=(m/s²)•m +(m/s²)•(m/s)
=> incorrect
2)s = s0 + kat^2
m= m + (m/s²)•s²=> correct
….
3 incorrect
4 correct
5 correct
Sure! Let's go through each equation and determine if they are dimensionally correct or incorrect. To do this, we need to ensure that the units on both sides of the equation are the same.
1) Equation: v^2 = 2as + kav
The left-hand side of the equation has units of (velocity)^2, while the right-hand side has units of (acceleration)(length) + (dimensionless)(acceleration)(velocity). Since both sides of the equation have the same units, this equation is dimensionally correct.
2) Equation: s = s0 + kat^2
The left-hand side has units of length, while the right-hand side has units of (length) + (acceleration)(time^2). These units do not match, so this equation is dimensionally incorrect.
3) Equation: a = v^2/r + kv/t
The left-hand side has units of acceleration, while the first term on the right-hand side has units of (velocity)^2/(length) and the second term has units of (dimensionless)(acceleration)(1/time). Since the units do not match, this equation is dimensionally incorrect.
4) Equation: t = v/a
Both sides of the equation have units of time, so this equation is dimensionally correct.
5) Equation: t = k√(s/g)
The left-hand side has units of time, while the right-hand side has units of (dimensionless) times the square root of (length)/(acceleration). Since the units do not match, this equation is dimensionally incorrect.
In summary, equations 1 and 4 are dimensionally correct, while equations 2, 3, and 5 are dimensionally incorrect.