Problem 74 - Calculus and its applications

Question

Find \(s(t)\). $$ v(t)=2 t, \quad s(0)=10 $$

Step-by-Step Solution

Answer

The position function is \( s(t) = t^2 + 10 \).

Step 1: Understand the Problem

We are given a velocity function \( v(t) = 2t \) and initial condition \( s(0) = 10 \). Our goal is to determine the position function \( s(t) \).

Step 2: Determine the Relationship Between Velocity and Position

The position function \( s(t) \) is the integral of the velocity function \( v(t) \) with respect to time \( t \). Thus, we must integrate \( v(t) \) to find \( s(t) \).

Step 3: Integrate the Velocity Function

To find \( s(t) \), integrate \( v(t) = 2t \):\[s(t) = \int 2t \, dt = t^2 + C\]where \( C \) is the constant of integration.

Step 4: Use Initial Conditions to Solve for the Constant

Substitute the initial condition \( s(0) = 10 \) into the equation to find \( C \).\[10 = 0^2 + C \ C = 10\]Thus, the constant \( C \) is 10.

Step 5: Write the Final Position Function

Substitute the constant \( C = 10 \) back into the position function:\[s(t) = t^2 + 10\]This is the position function that satisfies both the velocity equation and the initial condition.

Key Concepts

Velocity and Position Relationship

Velocity and position are closely connected in calculus. To find a position from a velocity function, we need to integrate.
This is because when we integrate a velocity function, we accumulate the changes in position to get the total position at any given time. If you imagine velocity as the speed of a car, then the position is how far the car has traveled.

• The velocity function tells us how fast the position changes - in this case, it's given by \( v(t) = 2t \).
• Integration helps us determine the accumulated distance or position over time.

In the given exercise, by integrating the velocity, we find the position function \( s(t) = t^2 + C \). This means the distance traveled at any time \( t \) is given by \( t^2 + C \).
Understanding this relationship is key to solving problems where you need to derive position from velocity.

Initial Conditions in Calculus

Initial conditions are specific values that allow us to solve for unknown constants in integration. In calculus, when we integrate to find a function like position from a derivative like velocity, we get a general solution. It includes a constant of integration \( C \), which can only be determined using initial conditions.

• Initial conditions provide information about the state of a system at a specific point, usually the starting point.
• In this problem, the initial condition \( s(0) = 10 \) tells us the position of the object at the time \( t=0 \).

By substituting different points into the position function \( s(t) = t^2 + C \), we use the initial condition to find that \( C = 10 \).
This ensures our integration solution fits the real-world scenario described.

Constant of Integration

Whenever you integrate a function, you often see a term called the constant of integration, or \( C \). This constant arises because integration can have multiple valid solutions differing by a constant amount.

• The constant of integration is crucial because it ensures all possible solutions are considered.
• Without it, certain particular solutions that fit specific scenarios would be impossible to determine.

In the problem above, after finding \( s(t) = t^2 + C \), the initial condition \( s(0) = 10 \) is used to determine \( C \). By substituting \( 0 \) for \( t \) and equating \( s(0) \) to \( 10 \), we solve for \( C \).
This teaches us the importance of \( C \) in making the solution specific to the conditions described in any given problem.