A particle follows the trajectory x(t)=(1)/(4) t^3+2 t, y(t)=20 t-t^2 with t in seconds and di

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A particle follows the trajectory x(t)=(1)/(4) t^3+2 t, ...

A particle follows the trajectory
$$
x(t)=\frac{1}{4} t^3+2 t, \quad y(t)=20 t-t^2
$$
with $t$ in seconds and distance in centimeters.
a. What is the particle's maximum height?
b. When does the particle hit the ground and how far from the origin does it land?

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Key Concepts

Parametric Equations

Parametric equations represent the coordinates of a moving object as functions of an independent parameter, typically time. This allows for a separate analysis of the horizontal and vertical motions, which is especially useful in two-dimensional motion problems where the path or trajectory of the object is not simply a straight line.

Differentiation for Finding Extrema

Differentiation is a central concept used to determine the rate of change of a function. In the context of finding a maximum height, differentiating the vertical component with respect to time and setting the derivative equal to zero helps identify the time at which the vertical displacement reaches a peak.

Solving Equations to Determine Key Events

Many problems in motion require finding the specific time when particular events occur, such as when an object returns to the ground. This typically involves solving equations (often quadratic) set up by equating the positional function (such as vertical displacement) to zero.

Horizontal Displacement Calculation

After determining the relevant time at which an event occurs (e.g., the particle hitting the ground), substituting this time into the horizontal position function provides the total horizontal distance traveled. This is essential for fully describing the trajectory or range of the projectile.

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