Math Problem Statement

An object moves in a one-dimensional motion starting at +4.5 m from the origin with a negative velocity of 2.1 m/s for 1.8 seconds. After that, it travels with a positive velocity of 6.3 m/s for 2 seconds. Immediately after that, it has a velocity equal to zero for 3.5 seconds. What is the final position of the object, expressed in meters to one decimal place?

Solution

We can break the problem down into three distinct stages of motion and calculate the displacement during each stage. We'll then add these displacements to the initial position of the object to find its final position.

Initial conditions:

  • Initial position: x0=+4.5mx_0 = +4.5 \, \text{m}
  • Time intervals:
    • Stage 1: t1=1.8st_1 = 1.8 \, \text{s}
    • Stage 2: t2=2.0st_2 = 2.0 \, \text{s}
    • Stage 3: t3=3.5st_3 = 3.5 \, \text{s}

Stage 1: Negative velocity

  • Velocity: v1=2.1m/sv_1 = -2.1 \, \text{m/s}
  • Displacement during Stage 1: Δx1=v1t1=(2.1)(1.8)=3.78m\Delta x_1 = v_1 \cdot t_1 = (-2.1) \cdot (1.8) = -3.78 \, \text{m}

Stage 2: Positive velocity

  • Velocity: v2=+6.3m/sv_2 = +6.3 \, \text{m/s}
  • Displacement during Stage 2: Δx2=v2t2=(6.3)(2.0)=12.6m\Delta x_2 = v_2 \cdot t_2 = (6.3) \cdot (2.0) = 12.6 \, \text{m}

Stage 3: Velocity equals zero

  • Velocity: v3=0m/sv_3 = 0 \, \text{m/s}
  • Displacement during Stage 3: Δx3=v3t3=0(3.5)=0m\Delta x_3 = v_3 \cdot t_3 = 0 \cdot (3.5) = 0 \, \text{m}

Final position calculation:

The total displacement is the sum of the displacements from each stage: Δxtotal=Δx1+Δx2+Δx3=3.78+12.6+0=8.82m\Delta x_{\text{total}} = \Delta x_1 + \Delta x_2 + \Delta x_3 = -3.78 + 12.6 + 0 = 8.82 \, \text{m}

The final position is the initial position plus the total displacement: xf=x0+Δxtotal=4.5+8.82=13.3mx_f = x_0 + \Delta x_{\text{total}} = 4.5 + 8.82 = 13.3 \, \text{m}

Final answer:

The final position of the object is 13.3 meters.

Would you like more details or have any questions?

Here are 5 questions related to this problem:

  1. How would the final position change if the initial velocity was positive instead of negative in Stage 1?
  2. What would be the total displacement if the object did not stop in Stage 3?
  3. How would air resistance affect the final position in this scenario?
  4. Could we calculate the average velocity of the object over the entire motion?
  5. How would the problem change if the object had a constant acceleration instead of constant velocity?

Tip: When dealing with motion problems, always break the motion into segments and calculate displacement for each separately.

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Math Problem Analysis

Mathematical Concepts

Kinematics
One-Dimensional Motion
Displacement
Velocity

Formulas

Displacement = velocity × time
Total displacement = sum of displacements
Final position = initial position + total displacement

Theorems

Equation of motion for constant velocity

Suitable Grade Level

Grades 9-12