projectile problem

Question

An object is projected horizontally from a measured vertical height and travels a measured horizontal distance. Design an experimental procedure by which the relationship between these values can be determined for a given launch speed. What do you predict the relationship to be? Make a detailed hypothesis!

Answer 1

You would like to show the relationship between height and range for different launch speeds. The variable that connects the x and y direction together is time - no matter what, both y and x stop at the same time (when the object stops).

In the x direction, the only equation we have is

V = x/t (velocity = distance over time)

The object is launched horizontally, so our launch speed is actually V. In the x direction, the speed of a projectile does not change, so this V will remain a constant. Therefore, the only expression we can make for x is:

X = V * t

In the y direction, we have a few equations to choose from:

Vf = Vi + at

Vf^2 = Vi^2 + 2ay

y = Vit + 1/2at^2

We know acceleration a = g = 9.8 and since the object was launched horizontally, we know the initial y velocity is 0

Vi = 0. We want y to be in our equation because we’re looking to find a relationship for it. We also want t to be in the equation because t connects x and y together. Let’s choose the last equation:

y = Vit + 1/2at^2

We can plug in what we know to simplify it. Vi = 0 and a = g (gravity)

y = 0 + 1/2gt^2

y = 1/2gt^2

Now we have an equation for y and an equation for x:

y = 1/2g * t^2

x = V * t

For a given launch speed, if we measure the amount of time the object is in the air, we can determine both the y and x distances. If we divide these equations with each other, we can see:

y/x = 1/2g *t^2 /Vt simplify. One of the t’s can cancel out and the 2 can be written with the denominator:

y/x = gt/2V

This tells us that if we graphed y and x, the slope of that line is gt/2V. We can also write it another way. Let’s get t/V by itself instead of y/x:

y/x = gt/2V multiply by 2 on both sides

2y/x = gt/V divide by g on both sides

2y/gx = t/V

This tells us that if we graphed our measured t and known V, the slope of that line is 2y/gx. We could be able to use our slope value to determine the relationship y/x by multiplying by g/2.

In summary, all we need to do for our experiment is measure the time at different given V launch speeds, then graph t on the y-axis and V on the x-axis. Then, we find the slope of the line using rise over run (algebra). This slope is equal to the rest of our equation, 2y/gx.

Slope of t/V line = 2y/gx then we simply get y/x by itself

slope * g / 2 = y/x

This is how we will find the relationship between y and x.

Hope this is what you were looking for!