Problem 52 - Calculus Single Variable

Question

This are concerned with a car's advance \(p(t)\) during its period of deceleration. Suppose that during the first \(t\) seconds of braking, the car moved forward \(p(t)=\) \(48 t-3 t^{2}-t^{3} / 2\) ft. Suppose also that this formula was in effect until the car came to a stop. What was the car's velocity at the moment brakes were applied?

Step-by-Step Solution

Answer

The car's velocity at the moment brakes were applied was 48 ft/s.

Step 1: Understanding the Problem

To find the car's velocity at the moment the brakes were applied, we need to differentiate the position function \( p(t) = 48t - 3t^2 - \frac{t^3}{2} \) with respect to time \( t \) to get the velocity function \( v(t) \). The velocity at \( t = 0 \) will give us the velocity when brakes were applied.

Step 1: Differentiate the Position Function

First, we differentiate the function \( p(t) = 48t - 3t^2 - \frac{t^3}{2} \) with respect to \( t \) to find the velocity function \( v(t) \).

Step 3: Computing the Derivative

Differentiate each term: - The derivative of \( 48t \) is \( 48 \),- The derivative of \( -3t^2 \) is \( -6t \),- The derivative of \( -\frac{t^3}{2} \) is \( -\frac{3t^2}{2} \).Thus, the velocity function is \( v(t) = 48 - 6t - \frac{3t^2}{2} \).

Step 2: Evaluate Velocity at \( t = 0 \)

Substitute \( t = 0 \) into the velocity function \( v(t) = 48 - 6t - \frac{3t^2}{2} \) to find the initial velocity when the brakes are applied.

Step 5: Calculation of Initial Velocity

Substitute \( t = 0 \) into the velocity function: \( v(0) = 48 - 6(0) - \frac{3(0)^2}{2} = 48 \).Thus, the car's initial velocity is \( 48 \) ft/s.

Key Concepts

Understanding Velocity Function

The concept of a velocity function relates closely to motion and speed. In physics, the velocity function represents how fast something moves at any given time. It describes the rate at which the position of an object changes over time. For a moving car, the velocity function tells us how quickly it covers distance at every instant.
To find it mathematically, usually, we differentiate the position function, which tells us the object's location over time. Differentiation helps us transition from position to velocity, giving a formula that reveals speed information for various time points.
Moreover, the velocity function can show us both positive and negative values. Positive values indicate that the object continues in the intended direction, whereas negative values suggest it moves backward or decelerates. Understanding the velocity function is crucial in analyzing motion dynamics.

Exploring Position Function

The position function is a mathematical representation highlighting an object's specific location related to time. For a car during deceleration, this function helps us understand where the car is at various moments as it slows down.
The expression of a position function often takes a polynomial form, like in our given example, which shows the path of the car during braking:

• First term: linear movement without acceleration,
• Second term: captures changes due to consistent acceleration or deceleration,
• Third term: higher-order changes echo further variation in speed.

The role of this function helps us transition smoothly into considering velocity by setting up the groundwork for differentiation.

Grasping Differentiation

Differentiation is a powerful mathematical tool helping us compute rates of change. When we differentiate a function, we find out how quickly its value changes as its variables alter.
For instance, when finding the velocity function from the position function, we use differentiation. This involves taking derivatives of each term in the position formula to capture how quickly the car’s position changes.

• From constant speed terms to changing speeds,
• to varying acceleration insights.

By deriving each part, we gain a new function, the velocity function, allowing us to analyze how speed evolves.
Differentiation offers a window to understanding movement's dynamics deeply, answering crucial questions about speed and how it shifts over time.

Understanding Initial Velocity

Initial velocity speaks to the beginning speed of an object or vehicle at a starting point. When brakes are applied to a car, understanding the initial velocity tells us how fast the car was going just before it began to decelerate. It's the velocity at the precise moment the brakes touch, often indicated by evaluating the velocity function at time zero.
In our problem, setting time equal to zero in the velocity formula reveals the car's starting speed. This intrinsic value lends insight into the car's motion context, from steady speeds to the point of its planned deceleration.

• Essential for safety calculations,
• Useful in planning stopping distances,
• Key in analyzing anticipated motion.

Initial velocity serves as a baseline for analyzing subsequent velocity changes, helping guide essential safety and performance assessments for vehicles.