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Sharon kicks a ball from the ground into the air with an upward velocity of 64 feet per second. The function $h=-16 t^2+64 t$ models the height $h$, in feet, of the ball at time $t$, in seconds. When will the ball reach the ground again?


A1 second after the ball is thrown
B2 seconds after the ball is thrown
C3 seconds after the ball is thrown
D4 seconds after the ball is thrown

Verified Answer :

  • Set the height equation to zero: \(-16t^2 + 64t = 0\).
  • Factor out the common term: \(-16t(t - 4) = 0\).
  • Solve for \(t\) by setting each factor to zero: \(t = 0\) or \(t = 4\).
  • The ball returns to the ground at \(t = 4\) seconds.
  • Final answer: \(\boxed{4}\) seconds (Option d).

Explanation

  1. Understanding the Problem

The problem asks to find the time \(t > 0\) at which a ball, kicked from the ground, returns to the ground. The height of the ball at time \(t\) is given by the function \(h(t) = -16t^2 + 64t\), where \(h\) is the height in feet and \(t\) is the time in seconds. When the ball is on the ground, its height is \(0\). Therefore, we need to solve the equation \(h(t) = 0\) for \(t > 0\).

  1. Setting up and Factoring the Equation

To find the time when the ball hits the ground, we set the height function equal to zero:
\[-16t^2 + 64t = 0\]
We can factor this quadratic expression by identifying the common factor \(-16t\):
\[-16t(t - 4) = 0\]

  1. Solving for Time and Identifying the Correct Answer

The equation \(-16t(t - 4) = 0\) holds if either factor is equal to zero. This gives us two possible times for when the height is zero:

  1. \(-16t = 0\), which gives \(t = 0\). This represents the moment the ball is initially kicked from the ground.
  2. \(t - 4 = 0\), which gives \(t = 4\). This represents the moment the ball returns to the ground after being in the air.

Since we are looking for the time the ball hits the ground again after being kicked, we choose \(t = 4\). This corresponds to option (d).

Examples

This model of a projectile's path is fundamental in physics and sports analysis. Coaches and athletes use such quadratic models to predict the trajectory of a ball, helping them determine the best angle and force to use for kicks, throws, or serves. By understanding when a ball will hit the ground, players can better anticipate where to position themselves for a catch or to defend against a play. For example, if a player kicks a ball with a specific initial velocity, they can calculate the exact time the ball will take to return, allowing them to optimize their performance and decision-making on the field.

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