Problem 54 - Essential Calculus

Question

Use \(a(t)=-32\) feet per second per second as the acceleration due to gravity. A balloon, rising vertically with a velocity of 8 feet per second, releases a sandbag at the instant it is 64 feet above the ground. (a) How many seconds after its release will the bag strike the ground? (b) At what velocity will the bag hit the ground?

Step-by-Step Solution

Answer

The sandbag hits the ground 2 seconds after release with a velocity of -56 feet per second.

Step 1: Establish The Parameters

First, establish the quantities: acceleration \(a=-32\)ft/s\(^2\) (an acceleration force downward), initial velocity \(v_0 = 8\)ft/s (the sandbag initially moves upward), and initial height \(s_0 = 64\)ft.

Step 2: Calculate Time

Now use the second equation of motion: \(s = s_0 + v_0t + (1/2)at^2\), where \(s\) is the final position (which is 0, as it's ground level). Substitute the given values, resulting in:\n0 = 64 + 8t - 16t^2.\nThis equation is a quadratic and can be solved by factoring. It simplifies to: \n0 = 4t^2 - 8t + 16.\nSolving for \(t\) gives two roots, but we discard the negative root (as time can't be negative), giving us \(t = 2\)s to satisfy the equation.

Step 3: Calculate Velocity

Next, use the first equation of motion, \(v = v_0 + at\), to calculate the final velocity. Substitute the known values, which results in: \(v = 8 - 32*2 = -56\)ft/s. The result is negative, which indicates the direction of the velocity is downward, as expected.

Key Concepts

Quadratic Equation

A quadratic equation is one of the fundamental tools in algebra, especially when solving problems related to projectile motion. It takes the form of \(ax^2 + bx + c = 0\). Here, the equation emerges when trying to determine when the sandbag hits the ground. By placing the final position, \(s\), at zero (ground level), we rearrange the equation of motion, \(s = s_0 + v_0t + \frac{1}{2}at^2\), into a quadratic form:

• \(a = -16\), due to half of the acceleration \(-32\) ft/s²,
• \(b = 8\), the initial velocity,
• \(c = 64\), the initial height.

These coefficients are then solved to find \(t\), the time it takes for the sandbag to reach the ground. Quadratics usually have two solutions. However, when dealing with projectile motion and time, only the positive solution makes sense.

Acceleration Due to Gravity

The acceleration due to gravity is a crucial concept in projectile motion, especially for objects moving vertically. It refers to the acceleration with which objects will experience an increase in velocity as they fall towards Earth due to the gravitational pull. In many problems related to projectile motion near the Earth's surface, it is commonly represented as \(-32\) ft/s², the negative sign indicating a downward direction.

• This value is consistent and helps predict how quickly an object's speed will change when moving under the sole influence of Earth’s gravity.
• This understanding allows us to predict the time it takes for the sandbag to hit the ground using a quadratic equation.

By knowing this acceleration, we can determine how fast an object picks up speed, translating it into calculations for velocity, as done in the solution.

Initial Velocity

Initial velocity describes the starting speed and direction of an object at the moment it begins its motion, marking its importance in motion equations. For the sandbag released from the balloon, its initial velocity is 8 ft/s upward. This initial motion component is vital because it directly affects how the object will travel before gravity alters its course.

• The positive value indicates an upward motion, opposite to gravity's pull.
• In the quadratic equation discussed, this initial velocity \(v_0\) serves as the \(b\) term, influencing time to impact.

Coupled with gravitational pull, the initial velocity's effect decreases over time until gravity maximizes its influence, completely reversing the sandbag's motion, thereby determining its descent.

Equation of Motion

Equation of motion is a central concept in solving problems related to moving objects. It combines the parameters of initial conditions and acceleration to forecast an object's location or velocity over time. The general form for linear motion is \(s = s_0 + v_0t + \frac{1}{2}at^2\), which allows us to connect position with time.

• \(s_0\) is the starting position,
• \(v_0\) is initial velocity,
• \(t\) is time,
• \(a\) is constant acceleration, such as gravity here.

In the given exercise, applying this equation lets us find how long the bag takes to fall. Additionally, velocity can be determined using a counterpart form, \(v = v_0 + at\), helpful in predicting how fast the sandbag hits the ground and anything about its path."