About This Quiz
The core concept tested in these questions is the application of quadratic functions to model projectile motion. A quadratic function is generally written as [latex]f(t) = at^2 + bt + c[/latex], where [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] are constants. In the context of projectile motion, the function models the height of an object over time.
The vertex of the parabola represented by the quadratic function provides critical information such as the maximum or minimum height. The vertex formula [latex]t = -\\frac{b}{2a}[/latex] helps determine the time at which the maximum or minimum height is achieved. Additionally, setting the function equal to zero ([latex]0 = at^2 + bt + c[/latex]) allows us to find the times at which the object is at ground level.
To successfully answer these questions, follow these steps:
- Identify the key components: Recognize the quadratic function provided in the problem and identify the coefficients [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex].
- Determine the vertex: Use the vertex formula [latex]t = -\\frac{b}{2a}[/latex] to find the time at which the maximum or minimum height is reached. Substitute this time back into the function to find the corresponding height.
- Solve for specific conditions: For questions asking when the object hits the ground, set the function equal to zero and solve the quadratic equation using the quadratic formula [latex]t = \\frac{-b \pm \\sqrt{b^2 - 4ac}}{2a}[/latex]. Choose the appropriate root based on the context of the problem.
- Check units and context: Ensure your answers are in the correct units and make sense within the context of the problem. For instance, negative times may not be physically meaningful in some scenarios.