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PHY 266
Projectile Motion and Numerical Integration.
I.. Trajectory with no air resistance
A. A canon ball is fired straight up with an initial velocity v
o
. Assuming no air
resistance, write an expression for the position of the ball as a function of time
y(t).
B. Now open the Excel file plotting the trajectory. Look at the different columns
and see how they are calculated. For one pair of columns, y and v are calculated
analytically using the equation from part a.
For the next pair, y and v are calculated numerically. This is done as follows.
First and initial position y(0) and velocity v(0) are defined. Knowing the net
force on the ball, the acceleration may be calculated using Newton’s 2
nd
law.
a(0) = F(0) / m
After a short interval of time dt, we can estimate the new velocity by assuming the
acceleration was constant over that interval.
v(1) = v(0) + a(0) * dt
Likewise, we can find the position at this time.
y(1) = y(0) + v(0) * dt
This process is then repeated over and over again to find later values of a, v and y:
a(n) = F(n) / m
v(n+1) = v(n) + a(n) * dt
y(n+1) = y(n) + v(n) * dt
Currently, the data only runs to 16 seconds. To extend the calculation, just copy and past
the last row.
This is numerical integration. (Specifically, this particular method is called a Left
Riemann Sum.) The process is illustrated below graphically. Note that as the interval dt
decreases, the accuracy of the estimate improves.
a(t)
Area =
∆
v
dt
