Math Problem Statement
Solution
To calculate the final velocity of the ball after it has traveled 10 meters, we use kinematics principles with the given graph.
Step 1: Understanding the graph
The graph shows a linear relationship between acceleration (m/s²) and distance (m).
From the graph:
- At , acceleration
- At , acceleration
Thus, the acceleration increases linearly as: where is the distance and is the constant of proportionality.
From the graph:
Therefore:
Step 2: Relating acceleration to velocity
The acceleration is not constant here; it depends on the distance. To find the velocity, we use the work-energy theorem (or integrate acceleration over distance).
The kinematic relationship between acceleration and velocity is: Given:
- (initial velocity),
- .
Substitute into the integral:
Step 3: Solving the integral
The integral of with respect to is:
Now evaluate the integral from to :
Thus:
Step 4: Calculating final velocity
Take the square root of :
Final Answer:
The final velocity of the ball after traveling 10 meters is approximately:
Do you want me to explain any part in more detail? Here are some related questions for further understanding:
- How do you derive from the graph?
- Why do we use the integral to calculate velocity when acceleration is not constant?
- How would the solution change if the acceleration were constant?
- What happens if the ball starts with an initial velocity other than ?
- How can we plot the velocity vs distance graph for this motion?
Tip: When acceleration varies with position, integrating acceleration helps determine the velocity at any point.
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Math Problem Analysis
Mathematical Concepts
Kinematics
Integral Calculus
Linear Acceleration
Formulas
a = k * d (acceleration as a function of distance)
v^2 = v_0^2 + 2 * ∫ a ds (kinematic relation between velocity and acceleration)
∫ 0.6d ds = 0.3d^2 (integrating acceleration with respect to distance)
Theorems
Work-Energy Theorem
Kinematic Equations
Suitable Grade Level
Grades 10-12
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