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Which equation would best help solve the following problem?



Hadley throws a baseball upward with an initial velocity of 12 meters per second. When he throws the ball, it is 2 meters above the ground. How long will it take the ball to hit the ground?


A$-4.9 t^2+12 t+2=0$
B$-16 t^2+12 t+2=0$
C$4.9 t^2+12 t+2=0$
D$-4.9 t^2+12 t=2$

Verified Answer :

  • Use the kinematic equation \(h(t) = -\frac{1}{2}gt^2 + v_0t + h_0\).
  • Substitute the values \(g \approx 9.8\), \(v_0 = 12\), and \(h_0 = 2\) into the equation to get \(-4.9t^2 + 12t + 2 = 0\).
  • Conclude that the equation describing the time \(t\) to hit the ground is \(\boxed{-4.9 t^2 + 12 t + 2 = 0}\).

Explanation

  1. Analyzing the Motion Equation

The problem describes a classic case of projectile motion under the influence of gravity. The height of an object at any time \(t\) can be modeled using the kinematic equation:
\[h(t) = -\frac{1}{2}gt^2 + v_0t + h_0\]
where:

  • \(g\) is the acceleration due to gravity (approximately \(9.8 \text{ m/s}^2\)).
  • \(v_0\) is the initial vertical velocity (in this case, \(12 \text{ m/s}\)).
  • \(h_0\) is the initial height above the ground (in this case, \(2 \text{ meters}\)).

The objective is to find the time \(t\) when the ball hits the ground, which corresponds to the point where the height \(h(t)\) becomes \(0\).

  1. Formulating the Equation for Time \(t\)

We substitute the known values into our kinematic equation:

  • Initial velocity \(v_0 = 12\)
  • Initial height \(h_0 = 2\)
  • Gravity \(g \approx 9.8 \text{ m/s}^2\), so \(\frac{1}{2}g \approx 4.9\)

Plugging these into the equation \(h(t) = -\frac{1}{2}gt^2 + v_0t + h_0\), we get:
\[h(t) = -4.9t^2 + 12t + 2\]
To find the time when the ball hits the ground, we set \(h(t) = 0\):
\[-4.9t^2 + 12t + 2 = 0\]

  1. Conclusion and Selection

Comparing the derived equation \(-4.9t^2 + 12t + 2 = 0\) with the given choices, it is evident that the first option is the correct one. The other choices involve either different units for gravity (like the \(-16\) coefficient often used with imperial units in feet/s²) or incorrect signs. Thus, the equation \(-4.9t^2 + 12t + 2 = 0\) allows us to solve for the time taken to hit the ground.

Examples

Understanding the physics of projectile motion is essential for fields ranging from sports science to engineering. For instance, athletes use these calculations to optimize their jump or throw distances, while game developers use similar kinematic equations to ensure that characters and objects in a game move realistically. By setting up the correct mathematical model, one can predict the precise moment or location where a moving object will land.

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