Three students derive the following equations in which x refers to distance traveled, v the speed, a the acceleration (m/s^2) , t the time, and the subscript zero (0) means a quantity at time t=0 (a) x=vt^2+2at (b) x=v_(0)t+(1)/(2)at^2 and (c) x=v_(0)t+2at^2 Which of these could possibly be correct according to a dimensional check?

Let's analyze each equation dimensionally to determine which one is possibly correct.<br /><br />(a) $x=vt^{2}+2at$<br /><br />- Distance traveled (x) has a dimension of length (L).<br />- Speed (v) has a dimension of length per unit time (L/T).<br />- Time (t) has a dimension of time (T).<br />- Acceleration (a) has a dimension of length per unit time squared (L/T^2).<br /><br />Substituting the dimensions into the equation:<br /><br />- $x = (L/T) \cdot (T)^2 + 2 \cdot (L/T^2) \cdot (T)^2$<br /><br />Simplifying:<br /><br />- $x = L + 2L = 3L$<br /><br />The dimensions on the right side do not match the dimension of length (L) on the left side. Therefore, equation (a) is not dimensionally correct.<br /><br />(b) $x=v_{0}t+\frac {1}{2}at^{2}$<br /><br />- Distance traveled (x) has a dimension of length (L).<br />- Initial speed ($v_{0}$) has a dimension of length per unit time (L/T).<br />- Time (t) has a dimension of time (T).<br />- Acceleration (a) has a dimension of length per unit time squared (L/T^2).<br /><br />Substituting the dimensions into the equation:<br /><br />- $x = (L/T) \cdot (T) + \frac{1}{2} \cdot (L/T^2) \cdot (T)^2$<br /><br />Simplifying:<br /><br />- $x = L + \frac{1}{2}L = \frac{3}{2}L$<br /><br />The dimensions on the right side match the dimension of length (L) on the left side. Therefore, equation (b) is dimensionally correct.<br /><br />(c) $x=v_{0}t+2at^{2}$<br /><br />- Distance traveled (x) has a dimension of length (L).<br />- Initial speed ($v_{0}$) has a dimension of length per unit time (L/T).<br />- Time (t) has a dimension of time (T).<br />- Acceleration (a) has a dimension of length per unit time squared (L/T^2).<br /><br />Substituting the dimensions into the equation:<br /><br />- $x = (L/T) \cdot (T) + 2 \cdot (L/T^2) \cdot (T)^2$<br /><br />Simplifying:<br /><br />- $x = L + 2L = 3L$<br /><br />The dimensions on the right side match the dimension of length (L) on the left side. Therefore, equation (c) is dimensionally correct.<br /><br />In conclusion, both equations (b) and (c) are dimensionally correct.