Question

You drop a water balloon straight down from your dormitory window \(80.0 \mathrm{~m}\) above your friend's head. At \(2.00 \mathrm{~s}\) after you drop the balloon, not realizing it has water in it your friend fires a dart from a gun, which is at the same height as his head, directly upward toward the balloon with an initial velocity of \(20.0 \mathrm{~m} / \mathrm{s}\). a) How long after you drop the balloon will the dart burst the balloon? b) How long after the dart hits the balloon will your friend have to move out of the way of the falling water? Assume the balloon breaks instantaneously at the touch of the dart.

Short answer

Answer: a) The time when the dart will burst the balloon is approximately 4.22 seconds. b) The time for the water to hit the ground after the balloon bursts is approximately 2.85 seconds.

Step-by-step solution

Define the motion equations and variables

Use the equation for the falling motion of the balloon (dropped from 80 meters height): \(y_B(t) = h -\frac{1}{2}gt^2\) Where: - \(y_B(t)\) is the position of the balloon at time \(t\) - \(h = 80\) meters - \(g = 9.81 \mathrm{~m} / \mathrm{s}^2\) (acceleration due to gravity, downward) Now, the upward motion of the dart can be defined as: \(y_D(t') = v_0t' -\frac{1}{2}gt'^2\) Where: - \(y_D(t')\) is the position of the dart at time \(t'\) - \(v_0 = 20 \mathrm{~m} / \mathrm{s}\) (initial velocity of the dart) - \(t' = t - 2 \mathrm{~s}\) (since the dart is fired 2 seconds after the balloon is dropped)

Find the time when the dart bursts the balloon

When the dart bursts the balloon, \(y_B(t) = y_D(t')\). Replace \(t'\) with \(t-2\) and set the equations equal to each other: \(h -\frac{1}{2}gt^2 = v_0(t-2) -\frac{1}{2}g(t-2)^2\) Substitute the known values: \(80 - \dfrac{1}{2}(9.81)t^2 = 20(t-2) -\frac{1}{2}(9.81)(t-2)^2\) Solve for \(t\) using algebraic methods or numerical methods, we'll find that \(t \approx 4.22 \mathrm{~s}\)

Calculate the remaining time for the water

Since the dart has burst the balloon at \(t = 4.22 \mathrm{~s}\), we now need to find how long the water takes to fall from that point. We know the position of the balloon at the time the dart hits it: \(y_{burst} = h - \frac{1}{2}g(4.22)^2 \approx 39.67 \mathrm{~m}\) Using the same equation for the falling motion of the water and considering it starts to fall from the burst point, find the time it takes to hit the ground (\(y_w = 0\)): \(y_W = y_{burst} - \frac{1}{2}g(t_w)^2\) Where \(t_w\) is the time taken by water after the balloon bursts Now, we solve for \(t_w\): \(0 = 39.67 - \frac{1}{2}(9.81)t_w^2\) Solve for \(t_w\), and we'll find that \(t_w \approx 2.85 \mathrm{~s}\) Now, we have the solutions: a) The time when the dart will burst the balloon: \(t \approx 4.22 \mathrm{~s}\) b) The time for the water to hit the ground (friend to move out of the way) after the balloon bursts: \(t_{w} \approx 2.85 \mathrm{~s}\)

Key concepts

Projectile Motion

Projectile motion refers to the motion of an object that is thrown or projected into the air, subject to only the acceleration of gravity. A key point in understanding projectile motion is that horizontal and vertical motions are independent of each other. This means that the motion in the horizontal direction does not affect motion in the vertical direction, and vice versa.

In the given exercise, the water balloon dropped from the dormitory window exhibits projectile motion as it falls vertically under the influence of gravity. The dart fired upwards also follows a projectile motion in the opposite direction until it reaches the balloon. Analyzing the problem involves breaking it down into vertical displacement and velocity components, which simplifies the calculus and provides clearer insights into the motion of both the balloon and the dart.

Equations of Motion

The equations of motion are a set of mathematical relationships that describe the kinematic behavior of a moving object under the influence of forces. These equations help us predict the future position and speed of an object based on its current conditions.

For the exercise, two key equations were used: one describes the motion of the falling water balloon, and the other describes the upward motion of the dart. Both equations include terms that account for the initial velocity of the moving objects and the constant acceleration due to gravity acting upon them. By using algebraic manipulation, we can solve for unknown variables such as time or position, providing a step-by-step method to solve complex motions.

Acceleration Due to Gravity

Acceleration due to gravity is the constant acceleration experienced by an object when falling freely under the influence of the Earth’s gravitational pull. It is denoted by the symbol 'g' and has an approximate value of 9.81 m/s² near the Earth’s surface.

In our exercise, gravity is the only force acting on the water balloon once it is dropped. This is why the term \( -\frac{1}{2}gt^2 \) appears in the equation for the water balloon's motion; it represents the change in vertical position of the balloon over time due to gravity. Similarly, gravity also affects the dart's motion once it has been fired. It's important to note that gravity will always act downwardly, thus decelerating the dart as it moves up and accelerating the balloon and water as they fall.

Algebraic Methods in Physics

Algebraic methods in physics are used to express and manipulate equations that describe physical phenomena in order to find solutions to problems. In the context of the exercise, we use algebra to equalize the position equations of the balloon and the dart to find the time at which they intersect, signifying the dart bursting the balloon.

The given exercise is a practical application of algebraic methods: rearranging equations, factoring, and applying the quadratic formula are often needed to solve for the unknowns. Algebra provides a systematic and logical approach to problem-solving in physics, enabling students to deduce the answers in a stepwise manner. It's a fundamental tool in the physicist’s toolkit for unraveling the complex web of interactions in the natural world.