Can someone explain the confusion with these basic motion equations?

[FlamingAero]

I have been reviewing the basic two-dimensional motion equations and I've discovered a conundrum that is causing me much confusion. For example, here is a basic formula with variables:

$v^2 = vi^2 + 2ax$

$v = ?$

$vi = 27$

$a = -7.5$

$x = 49$

Therefore:

$v^2 = 27^2 + 2(-7.5)(49)$

$v^2 = 729 + -735$

$v = √(-6)$

When I input the square root of (-6) into my calculator (a TI-83+), I receive a ERR:NONREAL ANS message. Are these values not compatible with this formula?

Here's another similar example, this time with the formula:

$ΔX = vi*t + (1/2)at^2$

$ΔX = 49$

$vi = 27$

$a = -7.5$

$t = ?$

I have no idea how to even arrange the equation in terms of $t$. Is this formula limited to solving displacement?

Thank you for your help and guidance.

 

[Doc Al]

Are these values not compatible with this formula?

Your values are just physically impossible. Given that initial velocity and acceleration, you'll never achieve x = 49. (Figure out the maximum value of x.)

Similar issue with the other formula for time. (In general, you can surely solve for the time. You'll get a quadratic equation.)

 

[FlamingAero]

Your values are just physically impossible.

I now see my error. The value $ΔX = 49$ was rounded for significant figures, and should have instead been $ΔX = 48.6$

Thank you for your help.

 

[FaraDazed]

Yeah the second one is a quadratic so you can either set it to 0 and factorise to get your two answers or use the quadratic formula below

$$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

 

[Nickie]

I can understand your confusion with these basic motion equations. It is important to note that these equations are derived from the laws of motion and are meant to describe the relationship between different variables involved in motion. However, there are certain assumptions and limitations to these equations that must be taken into consideration in order for them to be applicable.

For the first equation, v^2 = vi^2 + 2ax, it is important to note that this equation only applies to objects moving with constant acceleration. This means that the initial velocity (vi) and acceleration (a) must remain constant throughout the entire motion. In your example, the values of vi = 27 and a = -7.5 do not remain constant, as they are both changing with respect to time (t). This could be one reason why you are getting a non-real answer when solving for v. Additionally, taking the square root of a negative number is not possible in the real number system, which is why your calculator is giving you an error message.

For the second equation, ΔX = vi*t + (1/2)at^2, it is important to note that this equation only applies to objects moving with constant acceleration in a straight line. In your example, ΔX = 49 and vi = 27, but we do not have a value for t. This could be because the equation is not applicable in this scenario, as the object may not be moving in a straight line or may not have constant acceleration.

In conclusion, it is important to understand the assumptions and limitations of these equations in order to use them correctly. They are useful tools for describing the relationship between different variables in motion, but they may not always be applicable in all scenarios. I hope this explanation helps to clarify the confusion you were experiencing.