Math Problem Statement

Suppose an object moves along a line at 15 m/s, for 0 ≤ t < 2, and at 25 m/s, for 2 ≤ t ≤ 5, where t is measured in seconds. Sketch the graph of the velocity function and find the displacement of the object for 0 ≤ t ≤ 5.

Solution

The problem describes the velocity of an object moving along a line in two different time intervals, with the goal of sketching the velocity function and finding the displacement for 0≤t≤50 \leq t \leq 5.

Given Information:

  • Velocity = 15 m/s15 \, \text{m/s} for 0≤t<20 \leq t < 2
  • Velocity = 25 m/s25 \, \text{m/s} for 2≤t≤52 \leq t \leq 5

1. Sketching the velocity function:

The velocity function is piecewise constant:

\begin{cases} 15 \, \text{m/s} & \text{for } 0 \leq t < 2 \\ 25 \, \text{m/s} & \text{for } 2 \leq t \leq 5 \end{cases}$$ The graph would have a horizontal line at $$15$$ m/s from $$t = 0$$ to $$t = 2$$, and another horizontal line at $$25$$ m/s from $$t = 2$$ to $$t = 5$$. ### 2. **Finding the displacement:** Displacement is the integral of the velocity function over time, which is equivalent to the area under the velocity-time graph. We calculate the displacement separately for each interval: - **For $$0 \leq t < 2$$**: $$\text{Displacement} = v \times t = 15 \times (2 - 0) = 30 \, \text{m}$$ - **For $$2 \leq t \leq 5$$**: $$\text{Displacement} = v \times t = 25 \times (5 - 2) = 75 \, \text{m}$$ ### Total Displacement: $$\text{Total displacement} = 30 \, \text{m} + 75 \, \text{m} = 105 \, \text{m}$$ Thus, the total displacement over the interval $$0 \leq t \leq 5$$ is **105 meters**. Would you like further clarification or the graph of the velocity function? --- Here are 5 related questions for further exploration: 1. What is the difference between displacement and distance in this context? 2. How does a change in velocity affect the displacement over time? 3. Can you calculate the acceleration if the velocity were changing instead of constant? 4. How would you compute displacement if the velocity function were non-linear? 5. What is the significance of the slope of the velocity-time graph? **Tip**: When the velocity is constant, displacement is simply the product of velocity and time for that interval.

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Math Problem Analysis

Mathematical Concepts

Kinematics
Velocity
Displacement
Piecewise Functions

Formulas

Displacement = velocity × time
Piecewise constant velocity function

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 10-12

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