Problem 4 - Basic Technical Mathematics with Calculus

Question

A beach ball is rolled up a shallow slope with an initial velocity of \(18 \mathrm{ft} / \mathrm{s}\). If the acceleration of the ball is \(3.0 \mathrm{ft} / \mathrm{s}^{2}\) down the slope, find the velocity of the ball after \(8.0 \mathrm{s}\)

Step-by-Step Solution

Answer

The velocity of the ball after 8 seconds is \(-6 \text{ ft/s}\).

Step 1: Understand the Given Values

The initial velocity of the ball is given as \(v_0 = 18 \text{ ft/s}\). The acceleration \(a\) is \(-3.0 \text{ ft/s}^2\) due to it being against the direction of motion. We need to find the velocity \(v\) after \(t = 8.0 \text{ s}\).

Step 2: Use the Velocity Formula

We will use the formula for velocity with constant acceleration, which is given by \( v = v_0 + at \). In this equation, \(v_0\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time.

Step 3: Substitute the Known Values

Substitute the given values into the formula: \( v = 18 \text{ ft/s} + (-3.0 \text{ ft/s}^2) \times 8.0 \text{ s} \).

Step 4: Calculate the Final Velocity

Calculate the expression: \[ v = 18 \text{ ft/s} - 24 \text{ ft/s} = -6 \text{ ft/s} \]. This indicates that the ball has reversed direction after 8 seconds and is now moving down the slope.

Key Concepts

Understanding Velocity

Velocity is a fundamental concept in kinematics, describing how fast an object is moving and in which direction. It is a vector quantity, meaning it has both magnitude and direction. This differentiates it from speed, which only considers magnitude. For instance, if a beach ball is rolled uphill, its initial velocity might be given as 18 ft/s in a particular direction.

• Initial Velocity \(v_0\): This is the velocity of the object at the start of the observation or problem. In our example, the ball starts with a velocity of 18 ft/s.
• Final Velocity \(v\): This is the velocity of the object at a later time, often after some time has passed or some conditions have changed, like having acceleration act on it.

Calculating the velocity after a specific time when acceleration is acting can be done using the formula: \[ v = v_0 + a \times t \]This formula helps us understand how velocity changes over time with constant acceleration.

Exploring Acceleration

Acceleration is a measure of how quickly an object’s velocity changes over time. It is also a vector quantity, which means it has both magnitude and direction. Acceleration can cause an object to speed up, slow down, or change direction.

• Positive Acceleration: This usually results in the object increasing its speed.
• Negative Acceleration (Deceleration): Often denotes a decrease in speed, or acceleration in the opposite direction of motion, as in our example where the ball's acceleration is -3.0 ft/s².

The formula for acceleration is:\[ a = \frac{\Delta v}{\Delta t} \]Where \(\Delta v\) is the change in velocity, and \(\Delta t\) is the change in time. In the provided exercise, the negative acceleration indicates that the beach ball is slowing down in its initial direction and eventually starts moving in the opposite direction.

Introduction to Motion Equations

Motion equations are vital tools in kinematics that describe the relationship between an object's velocity, acceleration, time, and displacement. They allow us to predict the future position or velocity of an object when we have certain initial conditions.
The standard motion equation utilized here is:\[ v = v_0 + a \times t \]Here, the initial velocity \(v_0\), acceleration \(a\), and time \(t\) determine the final velocity \(v\). When considering more complex problems, other motion equations that include displacement such as:\[ s = v_0 t + \frac{1}{2} a t^2 \]can also be used to determine the position. These equations assume constant acceleration and are crucial in solving numerous motion-related problems.