The Parametric Tracer
Parametric equations and eliminating the parameter
Step 1 of 7: Watch the ball fly
A baseball is hit at 35 mph at a 50-degree angle. Use the time slider to watch it fly.
Drag the slider all the way to the right to see the full flight, then press Continue.
Step 2 of 7: Three views of one flight
Here are three graphs shown at once. Scrub the time slider and watch all three update together.
\(x(t)\) -- horizontal position
\(y(t)\) -- vertical position
Path \((x, y)\)
What shape is the \(x(t)\) graph? What shape is the \(y(t)\) graph?
Step 3 of 7: Where do these equations come from?
The ball is hit at 35 mph at 50 degrees. The horizontal component of velocity uses cosine, and the vertical component uses sine, just like in Simulation 7.
\(v_y = 35\sin(50°) \approx 26.81 \text{ ft/s}\)
So the position equations are:
\(y(t) = 35\sin(50°) \cdot t - 16t^2\)
The \(-16t^2\) is gravity pulling the ball down. The horizontal motion has no gravity, so it stays steady.
Which trig function controls the horizontal motion?
Step 4 of 7: Reading all three graphs together
Set the time slider to \(t = 1\) second and read the values from all three graphs.
\(x(t)\) -- horizontal
\(y(t)\) -- vertical
Path \((x, y)\)
Move the slider to approximately \(t = 1\). What is the ball's horizontal position at \(t = 1\)?
Step 5 of 7: Eliminate the parameter
We can remove \(t\) from the equations to get one equation in \(x\) and \(y\) alone.
From \(x(t) = 35\cos(50°) \cdot t\), solve for \(t\):
Substitute into \(y(t) = 35\sin(50°) \cdot t - 16t^2\):
That is a parabola in rectangular form. Watch it overlay the parametric path:
Step 6 of 7: What did you just lose?
Two balls travel the same parabolic arc. One is fast, one is slow. Watch them both.
Both balls have the same rectangular equation. What is different about them?
Step 7 of 7: Direction and speed
Compare three parametric equations that all trace a circle.
All three trace the same circle. What does the rectangular equation \(x^2 + y^2 = 1\) tell you about speed and direction?
Use the time slider above to watch the ball fly. The parametric equations describe where the ball is at each moment in time.
Explore Parametric Equations
Enter your own parametric equations or choose a preset. Use \(t\) as the parameter.
Could not parse equation. Use t as the variable. Supported: sin, cos, tan, sqrt, abs, pow, PI, E.
\(x(t)\)
\(y(t)\)
Path \((x, y)\)
Choose a preset or enter your own equations. Use the time slider or play button to trace the curve and see how x(t) and y(t) combine to form the path.
Apply: Parametric Equations
Basketball Free Throw
A basketball is released from a height of 7 feet, 15 feet from the hoop. The hoop is 10 feet high. The launch angle is 52 degrees.
The parametric equations are:
\(y(t) = 7 + v\sin(52°) \cdot t - 16t^2\)
Adjust the launch speed until the ball goes through the hoop. The ball makes it when it passes through \(x = 15\) at \(y = 10\) (within 0.5 feet).
What Shape Is This?
Trace the curve defined by \(x(t) = 3\cos t\), \(y(t) = 2\sin t\).
What shape is this curve?
Eliminate the parameter. What is the rectangular equation?
Did the Cars Crash?
Car A drives east along a highway: \(x_A = 30t\), \(y_A = 0\).
Car B drives north toward the same highway: \(x_B = 500\), \(y_B = 25t - 500\).
Time \(t\) is in seconds. Distances in feet.
Do the paths of the two cars cross?
Do the cars actually collide (same place at the same time)?
- Graph parametric equations by plotting points and tracing curves
- Eliminate the parameter to convert parametric equations to rectangular form
- Understand what information is preserved and lost when eliminating the parameter
- Distinguish between path (where) and trip (when, how fast, which direction)
- Apply parametric equations to projectile motion and other real-world contexts
Quick Check
A curve is traced by \(x(t) = 5t\), \(y(t) = 3t - 2\). After eliminating the parameter, the rectangular equation is:
Instructor Notes
Teaching Notes
This simulation builds on the trig decomposition from Simulation 7 (right triangle trig). The baseball example grounds parametric equations in physics students already know intuitively: things go sideways at one rate and up-then-down at another.
The three-graph display (x vs t, y vs t, and the xy-path) is the conceptual core. Students who can read all three together understand parametric equations. Those who cannot will struggle with everything that follows.
Steps 6 and 7 are the pedagogical payoff. The "what did you lose" step reveals that eliminating the parameter discards timing, and step 7 shows it also discards speed and direction. These are the facts students miss when they treat parameter elimination as an algebraic simplification rather than a trade.
Common Student Errors
- Treating intersecting paths as collisions. The challenge scenario directly tests this.
- Believing that eliminating the parameter is always desirable or "simpler." Steps 6-7 counter this.
- Forgetting to convert degrees to radians when computing trig values (especially in the baseball example).
- Confusing the x(t) graph (horizontal position vs time) with the actual path of the object.
Discussion Questions
- A GPS tracker records your position every second. Is it giving you parametric data or rectangular data? What could you figure out from it that a map of your route could not?
- Two runners complete the same 400m oval track. Their paths are identical. What distinguishes a 50-second lap from a 70-second lap, and which representation captures that difference?
- When would you prefer the rectangular form over the parametric form? When would parametric be better?
Exam Connection
Expect questions that ask students to (1) eliminate the parameter from a pair of parametric equations, (2) identify the resulting conic, and (3) explain what information the parametric form carries that the rectangular form does not. The car-collision scenario mirrors a common exam format: "do the paths intersect?" vs "do the objects meet?"