Problem 38 - Calculus Single Variable

Question

A velocity \(v(t)\) is given. Calculate the acceleration (in \(\mathrm{m} / \mathrm{s}^{2}\) ). \(v(t)=t^{2}-5 t \mathrm{~m} / \mathrm{s}\)

Step-by-Step Solution

Answer

The acceleration is \(a(t) = 2t - 5\) \(\text{m/s}^2\).

Step 1: Identify the relationship

Acceleration is the derivative of velocity with respect to time. Given the velocity function is \(v(t) = t^2 - 5t\), we need to find its derivative \(v'(t)\).

Step 2: Differentiate the velocity function

To find the acceleration \(a(t)\), differentiate the velocity function \(v(t) = t^2 - 5t\) with respect to \(t\). The derivative of \(t^2\) is \(2t\), and the derivative of \(-5t\) is \(-5\).

Step 3: Write the expression for acceleration

Combine the derivatives to express the acceleration function: \(a(t) = 2t - 5\). This is the acceleration as a function of time in \(\text{m/s}^2\).

Key Concepts

Understanding Calculus

Calculus is a branch of mathematics that primarily deals with change and motion. It offers revolutionary concepts like differentiation and integration, which are essential to understand processes that involve change. In this context, we use calculus to find how a quantity evolves over time.

Differentiation, in particular, is the aspect of calculus we are mostly concerned with in this exercise. Differentiation helps us determine the rate at which one quantity changes with respect to another. In simpler terms, it tells us how a function's output changes as its input changes.

When discussing motion, differentiation allows us to compute how velocity changes over time, which is crucial in finding acceleration. All these calculations are needed to analyze the dynamics of moving objects thoroughly.

Unpacking Velocity in Motion

Velocity is a vector quantity that refers to the speed of something in a particular direction. Unlike speed, which is scalar, velocity incorporates both magnitude and direction. In this exercise, velocity is given as a function of time.

• Velocity function: Describes how velocity changes with respect to time.
• Importance: Knowing the velocity function helps predict how fast and in what direction an object is moving at any given time.

Think of the velocity function as telling a story of motion. It reveals how quickly an object travels and in what direction, transforming raw numbers into a vivid narrative about movement. Typically, the velocity changes smoothly with time, making calculus the perfect tool to understand these changes.

Decoding Acceleration

Acceleration is another vector quantity, indicating the rate at which an object's velocity changes over time. It is crucial in understanding any object's dynamics because it provides insights into how quickly an object is speeding up or slowing down.

Acceleration results from the derivative of the velocity function. In simple terms, it is the rate of change of velocity with respect to time. If velocity tells us the movement story, then acceleration tells us how that story is evolving.

• Calculation: Differentiate the velocity function to find acceleration.
• Expression: The derivative gives an acceleration function.
• Units: Acceleration is often expressed in meters per second squared (\(\text{m/s}^2\)).

For example, in our case, differentiating \(v(t) = t^2 - 5t\) gives an acceleration function \(a(t) = 2t - 5\), which explicitly describes how acceleration varies with time.