The graph of kinetic energy of a body against the square of its velocity is shown in the figure below. What is the momentum of the body when its velocity is 2m/s?
Understand the Problem
The question asks to calculate the momentum of a body given a graph of its kinetic energy versus the square of its velocity and a specific velocity value. We need to determine the mass of the object from the slope of the graph and then use the mass and given velocity to calculate the momentum.
Answer
$16 \text{ kg m/s}$
Answer for screen readers
$16 \text{ kg m/s}$
Steps to Solve
- Relate Kinetic Energy and Velocity
We know that the kinetic energy (KE) is related to velocity ($v$) and mass ($m$) by the formula: $KE = \frac{1}{2}mv^2$
- Express KE in terms of $v^2$
The problem provides a graph of KE vs $v^2$. We can rewrite the KE equation to highlight the relationship between KE and $v^2$: $KE = (\frac{1}{2}m)v^2$ This equation resembles the form $y = mx$, where $y = KE$, $x = v^2$, and the slope is $\frac{1}{2}m$.
- Determine the slope of the graph
From the graph, the slope is given as 2. Therefore: $\frac{1}{2}m = 2$
- Solve for mass (m)
Multiply both sides of the equation by 2: $m = 4 \text{ kg}$
- Calculate Momentum
Momentum ($p$) is given by the formula: $p = mv$ We are given that $v=4 \text{ m/s}$, and we calculated $m = 4 \text{ kg}$. Therefore: $p = (4 \text{ kg})(4 \text{ m/s})$
- State the final answer
$p = 16 \text{ kg m/s}$
$16 \text{ kg m/s}$
More Information
Momentum is a vector quantity, possessing both magnitude and direction. In this problem, we're only calculating the magnitude of the momentum.
Tips
A common mistake is to forget the factor of $\frac{1}{2}$ in the kinetic energy formula or to confuse kinetic energy with momentum. Additionally, students may incorrectly read the slope from the graph, or make algebraic mistakes when solving for the mass.
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