Problem 44 - Precalculus

Question

Involve vertical motion and the effect of gravity on an object. Because of gravity, an object that is projected upward will cventually reach a maximum height and then fall to the ground. The equation that determines the height \(h\) of a projectile \(t\) seconds after it is shot upward is given by $$ h=\frac{1}{2} a t^{2}+v_{0} t+h_{0} $$ where \(a\) is the acceleration due to gravity, \(h_{0}\) is the initial height of the object at time \(t=0,\) and \(v_{0}\) is the initial velocity of the object at time \(t=0 .\) Note that a projectile follows the path of a parabola opening down, so \(a<0\) Can the human brain perform more calculations per second than a supercomputer? The calculating speed of the three top supercomputers, IBM's Blue Gene/L, IBM's BGW, and IBM's ASC Purple, has been determined. The speed of IBM's Blue GenelL is 245 teraflops more than that of IBM's BGW. The computing speed of IBM's BGW is 22 teraflops more than that of IBM's ASC Purple. The combined speed of all three top supercomputers is 568 teraflops. Determine the computing speed (in teraflops) of each supercomputer. A teraflop is a measure of a computer's speed and can be expressed as 1 trillion floating-point operations per second. By comparison, it is estimated that the human brain can perform 10 quadrillion calculations per second.

Step-by-Step Solution

Answer

ASC Purple: 93 teraflops, BGW: 115 teraflops, Blue Gene/L: 360 teraflops.

Step 1: Understand the Problem

We need to find the computing speed of each of the three supercomputers in teraflops given that their combined speed is 568 teraflops. The speeds of the supercomputers are related to each other.

Step 2: Define the Variables

Let's denote the computing speed of IBM's ASC Purple as \( x \) teraflops. Then IBM's BGW would be \( x + 22 \) teraflops (since it's 22 teraflops more than ASC Purple), and IBM's Blue Gene/L would be \( x + 22 + 245 \) teraflops (since it's an additional 245 teraflops more than BGW).

Step 3: Set Up the Equation

Based on the total computing speed of 568 teraflops, we set up the equation: \[ x + (x + 22) + (x + 22 + 245) = 568 \]

Step 4: Simplify the Equation

Combine like terms in the equation: \[ x + x + 22 + x + 22 + 245 = 568 \] which simplifies to \[ 3x + 289 = 568 \]

Step 5: Solve for x

To solve for \( x \), subtract 289 from both sides: \[ 3x = 568 - 289 \] which gives \[ 3x = 279 \] Then, divide both sides by 3 to isolate \( x \): \[ x = \frac{279}{3} = 93 \]

Step 6: Find Each Supercomputer's Speed

Now that \( x = 93 \), compute the speed of each supercomputer:1. IBM's ASC Purple: \( x = 93 \) teraflops.2. IBM's BGW: \( x + 22 = 93 + 22 = 115 \) teraflops.3. IBM's Blue Gene/L: \( x + 22 + 245 = 115 + 245 = 360 \) teraflops.

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and is influenced primarily by its initial velocity and gravity. When an object is projected, it follows a curved path. This path, known as a trajectory, is shaped by the forces acting on the object, including its initial velocity and air resistance. However, in a vacuum or ideal conditions where air resistance is negligible, the primary force acting upon a projectile is gravity.

Key aspects to consider in projectile motion include:

• Initial velocity: the speed at which the object is launched.
• Angle of launch: if an object is launched at an angle, it has both horizontal and vertical components of velocity.
• Gravity: it affects the vertical motion of the projectile, pulling it downward.

Due to the effect of gravity, an object that has been projected will reach a maximum height and then descend. This is a crucial factor that differentiates projectile motion from other types of motion.
Understanding how these factors work together can help predict the object's path, speed at varying points, and the maximum height reached by a projectile.

Acceleration Due to Gravity

Acceleration due to gravity, often denoted by the symbol \( g \), is the acceleration of an object caused by the force of gravity acting upon it. On Earth, this acceleration is uniformly approximated as \( 9.8 \text{ m/s}^2 \) downward.

When dealing with vertical or projectile motion, gravity is the force that changes the projectile's path by continuously accelerating it downward. In standard physics problems, it's common to treat gravity as having a constant value of \( -9.8 \text{ m/s}^2 \). This negative sign indicates that gravity acts in the downward, or negative, direction with respect to the initial launch direction.

The effects of gravity on a projectile are predictable in several ways:

• It slows down the projectile's upward motion until it stops.
• It then reverses the projectile, causing it to accelerate downwards.
• The constant gravitational pull ensures that the trajectory is symmetrical: the path upwards is a mirror image of the path downwards if starting and ending at the same vertical level.

Recognizing the influence of gravity allows us to estimate various parameters of motion: the time it takes for an object to reach its highest point, the total duration in the air, and the impact velocity upon returning to its original height.

Parabola Trajectory

The parabolic trajectory is the path that a projectile follows when launched into the air. This path resembles the shape of a parabola, which is a specific type of symmetrical curve. In projectile motion, the trajectory is crafted by the combination of initial upward velocity and the consistent downward acceleration due to gravity.

This parabolic nature can be described mathematically using a quadratic function of the form \( h(t) = -\frac{1}{2} g t^2 + v_0 t + h_0 \):

• \( t \) represents the time variable, indicating the passage of time after launch.
• \( h(t) \) represents the varying height of the projectile at different times.
• The term \( -\frac{1}{2} g t^2 \) reflects the influence of gravity, which causes the values of \( h(t) \) to decrease quadratically over time.
• The term \( v_0 t \) accounts for the initial velocity influence over time.
• \( h_0 \) stands for the initial height from which the projectile was launched.

Understanding the parabolic trajectory of a projectile helps in making accurate predictions about its position at any given time, ensuring precise calculations are made for impacts in sports, engineering, and space missions. With this knowledge, knowing exactly when and where an object will land becomes a calculable certainty.