# Question #ec930

Feb 8, 2015

The answer to this problem is 3.2 meters (using two significant figures if necessary).

When you do projectile problems (and others involving motion, like static or dynamic problems), you always have to first draw a picture. This will help you get to use which necessary equations and variables you need to solve for.

Below here is an example of how I've drawn the scenario:

In this diagram:

• I state the coordinate system to set my directions for velocity, acceleration, and displacement.
• I drew the picture of a dart thrown horizontally at some initial velocity (in x-direction) and other variables including acceleration acting down (due to gravity), displacements, and the time frames.
• Finally, I state the givens and unknowns (easy to follow and to plug in the numbers). We know the y-displacement to be -0.32 m (negative because of our coordinate system), y-acceleration to be -9.8 m/${s}^{2}$ (gravity), and velocity in the x-direction going constantly at +12.4 m/s. There is no acceleration in the x-direction.

Once I got the information ready to solve, I find the equations to use. We know that we must use the equation

(1) ${v}_{x} = \frac{\Delta x}{\Delta t}$

to solve for the displacement in the x-direction. We do not have the change in time (assuming the initial time is 0 s), but we do know the acceleration in the y-direction. So we can use the equation

(2) $\Delta y = \frac{1}{2} {a}_{y} {\left(\Delta t\right)}^{2} + {v}_{i , y} \left(\Delta t\right)$,

with the notion that the initial velocity in the y-direction is 0 m/s.

Cool, now we got our "tools." So what I will do next is set the first equation to $\Delta t = \frac{\Delta x}{v} _ x$ and substitute it for the second equation. Time is interchangeable between different directions.

Eventually, with ${v}_{i , y} \left(\Delta t\right)$ canceled to zero, you will get the equation

(3) $\Delta y = \frac{1}{2} {a}_{y} {\left(\frac{\Delta x}{v} _ x\right)}^{2}$

With this, you can solve for $\Delta x$ as a variable and plug in the numbers to get the answer. Make note of the signs from our givens and by the coordinate system.

Notice how I solve for the equations using variables and then plug in the answers. It is more efficient to do so in limited time.

Be aware that some projectile problems may include objects thrown at certain angles. You have to define the x and y components of velocity using cosine or sin trig functions (usually, it helps to draw a triangle in those cases).

For other problems, make sure to follow a similar step-by-step procedure by following the key components in the problem and you will be able to understand how to get the answer. That is what I am doing with such a long text (sorry about that). Hopefully all of these help.