Physics Lab #6

 

Projectile Motion II

Theory

For an object traveling in earth's gravity in such a way that we can ignore air resistance, its horizontal (x) motion and vertical (y) motion is described by:

 

For position:

x = v_ocosθ t,

y = v_osinθ t – ½g t²,

 

For velocity

v_x = v_ocosθ,

v_y = v_osinθ –  g t

 

For acceleration:

a_x = 0 m/s²,

a_y = – g = – 9.8 m/s²

 

where the initial position and time are chosen to be zero and the initial velocity is of magnitude v_o (also known as launch speed) directed at angle θ above the horizontal.  The horizontal direction is x and the vertical direction is y (up positive, down negative).

 

Using these equations we can show that the maximum height, H, the range (distance traveled horizontally if it begins and ends at the same height, i.e. the ground is flat), R, and the total time in the air, T are given by:

 

H= ½v²_osin²θ/g,

R= v²_osin2θ/g,

T= 2v_osinθ/g

 

Remember from the last funnelator lab:
From trigonometry we can figure out how tall something is, call it H, if we know how far we are from the base of the object, call it d, and we measure the angle, A, from the base to the top of the object. We get:
H = d tan(A)

Procedure

We will use the funnelator to shoot a rubber ball upward in a parabolic trajectory.  (Two bonus marks for those who can use the first two equations in the theory section to prove the shape is a parabola). We will use trigonometry to figure out the height of the ball at maximum and then time the duration of the entire trip, from start until it hits the ground. We will do a number of launches. Remember reading errors.

1. We will do a few practice runs to figure out the approximate highest point of the path.  We will set up the sighting group about 20 – 25 m out from this point. (Measure and note this distance with error.  It is d.)
2. The group at the launch site will try to get a good measurement of the angle of launch, θ.
3. When the ball is launched one partner should sight along a meter stick and hold the stick in place where the maximum height occurs. Measure and note the angle the meter stick makes with the ground. This is angle A used with d to get the height, H.
4. The timer should start the timer when the ball is launched and stop it upon landing.
5. A final group will measure out the range of the launch, so someone needs to note where the ball lands and use the tape measure to find the distance from launch point to landing point.
6. We will try to get 2 or 3 launches in.
7. After that we will try to show that the angle for the longest range is 45º if we keep the launch speed constant.  To keep the launch speed constant we need to keep the amount the funnelator is pulled back constant.  Someone also needs to make note of the launch angle. We will try this for a range of angles.

Calculations

We will assume that our launch angles are accurate and then calculate the launch speed from the H, R and T measurements.  Compare results.  Do they agree?  Which is the most precise?  From the first projectile lab, what misgivings do you have about all this?  How might you fix these results?  Also try to check whether or not the maximum range was found for the expected angle.