Acceleration to position Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.


a(t) = -32; v(0) = 20, s(0) = 0

Verified step by step guidance

1. Start by recalling the relationship between acceleration, velocity, and position. Acceleration is the derivative of velocity, and velocity is the derivative of position. To find the position function, we need to integrate the acceleration function twice.
2. Integrate the acceleration function a(t) = -32 with respect to time t to find the velocity function v(t). This gives v(t) = ∫(-32) dt = -32t + C₁, where C₁ is the constant of integration.
3. Use the initial condition v(0) = 20 to solve for C₁. Substitute t = 0 and v(0) = 20 into the velocity equation: 20 = -32(0) + C₁. Solve for C₁.
4. Now integrate the velocity function v(t) = -32t + C₁ with respect to time t to find the position function s(t). This gives s(t) = ∫(-32t + C₁) dt = -16t² + C₁t + C₂, where C₂ is another constant of integration.
5. Use the initial condition s(0) = 0 to solve for C₂. Substitute t = 0 and s(0) = 0 into the position equation: 0 = -16(0)² + C₁(0) + C₂. Solve for C₂. Once C₁ and C₂ are determined, substitute them back into the position function s(t) to get the final expression.

Key Concepts

Acceleration

Acceleration is the rate of change of velocity with respect to time. In this context, the acceleration function a(t) = -32 indicates a constant acceleration, which means the object's velocity decreases uniformly over time. Understanding acceleration is crucial for determining how it affects the object's velocity and position as it moves along a line.

Velocity

Velocity is the integral of acceleration and represents the rate of change of position with respect to time. Given the initial velocity v(0) = 20, we can find the velocity function by integrating the acceleration function. This step is essential for determining how the object's speed and direction change over time, which directly influences its position.

Position Function

The position function describes the location of an object at any given time and is obtained by integrating the velocity function. With the initial position s(0) = 0, we can establish the position function by integrating the velocity function derived from the acceleration. This concept is vital for understanding the object's trajectory and final position after a certain time.