Question

Free Fall A water balloon dropped from a window high above the ground falls \(y = 4.9 t ^ { 2 } \mathrm { m }\) in \(t\) sec. Find the balloon's (a) average speed during the first 3 sec of fall. \( (b) speed at the instant \)t = 3$

Short answer

a) The average speed of the water balloon during the first 3 sec of fall is \(14.7 m/s.\) b) The speed of the balloon at the instant \(t = 3\) is \(29.4 m/s.\)

Step-by-step solution

Calculate the total distance

From the provided equation, we have \(y = 4.9t^2\) that gives us the distance traveled in a given time. Using this formula, we plug in t = 3 sec to find out how far the balloon has fallen in the first 3 seconds which is \(y = 4.9 * (3)^2 = 44.1 m \)

Find the average speed

Average speed is total distance divided by total time. Having found the total distance y traveled in 3 sec in the previous step, we now divide it by the total time (3 sec). The average speed, \(v_{avg}\), is thus \(v_{avg} = \frac{44.1 m}{3 sec}= 14.7 m/s.\)

Calculation of instantaneous speed at \(t = 3\)

The instantaneous speed, or velocity, at a given time is the derivative of the displacement with respect to time. Here, we differentiate \(y = 4.9t^2\) with respect to \(t\) to find the velocity, \(v = dy/dt = 2*4.9*t = 9.8t\). Substituting \(t = 3\) into this formula, the instantaneous speed at \(t = 3\) is \(v = 9.8*3= 29.4 m/s.\)

Key concepts

Average Speed Calculation

When studying motion, average speed is a fundamental concept that helps us understand how fast an object is moving over a period of time. To calculate the average speed, you take the total distance traveled and divide it by the total time taken to travel that distance. The formula is simply:
\[ v_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} \]
In the context of free fall, like the exercise with the water balloon, we can use the given position-time equation to determine the distance fallen after a certain time. After calculating this distance, we can apply the formula for average speed by dividing the distance by the time for the journey. In our exercise, the water balloon fell 44.1 meters in 3 seconds, so its average speed was
\[ v_{avg} = \frac{44.1 \text{ m}}{3 \text{ sec}} \approx 14.7 \text{ m/s} \].
This calculation gives a simple yet powerful understanding of how fast the balloon was moving on average during its fall.

Instantaneous Speed

While average speed gives us a general idea of motion over a span of time, instantaneous speed tells us how fast an object is moving at a specific moment. It is equivalent to the magnitude of the instantaneous velocity. For a more precise understanding of the concept, think of the speedometer in a car that shows the speed at any given instance.
To calculate instantaneous speed in free fall, or any motion, you take the derivative of the distance with respect to time. If the position is described by a function of time, such as \( y(t) \), then the instantaneous speed is the first derivative of that function, noted as \( v(t) = \frac{dy}{dt} \).
In our exercise's scenario, the derivative of the position function \( y = 4.9t^2 \) is \( v = 9.8t \), representing the instantaneous speed. At \( t = 3 \) seconds, the instantaneous speed for the water balloon is \( v = 9.8 \times 3 = 29.4 \text{ m/s} \), which indicates just how fast the balloon was moving at that very second.

Kinematic Equations

To thoroughly understand free fall and the motion of objects under the influence of gravity, we must delve into kinematic equations. These equations describe the relationships among time, displacement, velocity, and acceleration under constant acceleration, such as gravity. There are several kinematic formulas, but one of the most used, and relevant to our exercise, is:
\[ y = vt + \frac{1}{2}at^2 \]
Here, \( y \) represents the displacement, \( v \) is the initial velocity, \( a \) is the acceleration (in free fall, this is due to gravity), and \( t \) is the time.
Since the water balloon is dropped, the initial velocity is 0, and the only acceleration is from gravity (which is approximately \( 9.8 \text{ m/s}^2 \) on Earth), simplifying our equation to \( y = \frac{1}{2}at^2 \). The exercise provided uses \( y = 4.9t^2 \), which corresponds to \( 4.9 \) being half of gravity's acceleration.
By understanding kinematic equations, one can analyze various aspects of motion, predict future positions, and calculate velocities. It's vital for students to familiarize themselves with these equations for a comprehensive grasp of motion in physics.