91Ó°ÊÓ

Motion equations are a set of equations used to describe an object's motion in terms of its position, velocity, acceleration, and time. In our problem with the compass falling from a hot-air balloon, we used the equation:
\[ h = v_i t + \frac{1}{2} a t^2 \]
Here:

• \( h \) represents the height or displacement, which in this case is the initial height of 3.00 meters from which the compass is dropped.
• \( v_i \) is the initial velocity of the compass. Since the balloon and the compass are moving upward initially, \( v_i = 2.50 \text{ m/s} \).
• \( a \) is the acceleration, which is due to gravity acting in the opposite direction to the initial velocity, thus it is \(-9.81 \text{ m/s}^2\).
• \( t \) is the time period over which this motion occurs, which is what we want to solve for.

This equation is ideal for objects in linear motion with constant acceleration, such as freely falling bodies under the influence of gravity.

The quadratic formula is a powerful tool in mathematics used to find the roots of quadratic equations. A quadratic equation has the form:
\[ ax^2 + bx + c = 0 \]
In our example, after substituting in the given values and simplifying, we ended up with the quadratic equation:
\[ 4.905t^2 - 2.50t - 3.00 = 0 \]
To solve for \( t \), we use:

• \( a = 4.905 \)
• \( b = -2.50 \)
• \( c = -3.00 \)

The quadratic formula is:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
By calculating the discriminant \( b^2 - 4ac = 65.11 \), it shows us how many real solutions the equation has. Here, it’s positive, meaning we have two real solutions.
Finally, substituting into the quadratic formula, we find the positive time solution \( t \approx 1.07 \) seconds, which is realistic since time can't be negative. This tells us how long it takes for the compass to reach the ground.

Acceleration due to gravity is a crucial concept when studying motion, specifically free-fall scenarios. On Earth, the standard value for this acceleration is \( 9.81 \text{ m/s}^2 \) and acts downwards towards the center of the Earth. In our problem, the compass experiences this gravitational pull once released from the balloon.
When analyzing kinematics problems, gravity usually acts as a constant acceleration \( a \). Since our compass initially moves upwards due to its initial velocity from the balloon, gravitational acceleration \(-9.81 \text{ m/s}^2\) acts against this motion.
Combining initial upward velocity with downward acceleration, we calculate time until the object ceases upward motion and starts moving downward, ultimately falling to the ground.

• It influences how fast the velocity of the object increases downwards as time progresses.
• In expressions like \( v = u + at \), it helps determine how quickly the motion shifts due to forces acting upon it.

Understanding acceleration due to gravity is essential for accurate kinematics calculations, enabling us to predict the behavior of free-falling objects accurately.