Question

A 3.0 kg object moving at 8.0 m/sin the positive direction of an xaxis has a one-dimensional elastic collision with an object of mass M, initially at rest. After the collision the object of mass Mhas a velocity of6.0 m/sin the positive direction of the axis. What is mass M?

Short answer

Mass of the object, M = 5.0 kg .

Step-by-step solution

Understanding the given information

1. Mass, m = 3.0 kg .
2. The velocity of mass m,$v_{i1}=8.0\mathrm{m}/\mathrm{s}$.
3. The velocity of mass M, $v_{f2}=6.0\mathrm{m}/\mathrm{s}$.

 Step 2: Concept and formula used in the given question

You use the concept of conservation of momentum and conservation of kinetic energy. You can write the two equations, one for conservation of momentum and another for conservation of kinetic energy. You will have two unknowns; you can rearrange one equation for one of the unknowns, and by plugging it into another equation, you will get an equation with only one unknown, and then you can solve for that unknown.

Calculation for the mass M

We know the initial velocity of mass M is $v_{i2}=0$, and the final velocity of mass m$v_{f1}$is unknown.

From the conservation of momentum principle, we can write,

$m_1{v}_{i1}+m_2{v}_{i2}=m_1{v}_{f1}+m_2{v}_{f2}$

We plug the values in the equation of conservation of momentum, and we get,

$$\begin{gathered} \left(3.0\right)\left(8.0\right)+M\left(0\right)=\left(3.0\right)v_{f1}+M(6.0) \\ 24=\left(3.0\right)v_{f1}+\left(6.0\right)M \end{gathered}$$

We can rearrange this equation for, andwe get,

$$\begin{gathered} 24-\left(6.0\right)M=\left(3.0\right)v_{f1} \\ {v}_{f1}=8-2M \end{gathered}$$ (1)

From the kinetic energy principle, we can write,

$\left(\frac{1}{2}\right)m_1{\left(v_{i1}\right)}^2+\left(\frac{1}{2}\right)m_2{\left(v_{i2}\right)}^2=\left(\frac{1}{2}\right)m_1{\left(v_{f1}\right)}^2+\left(\frac{1}{2}\right)m_2{\left(v_{f2}\right)}^2$

Plugging the values in the equation of conservation of kinetic energy, we get,

$\left(\frac{1}{2}\right)\left(3.0\right){\left(8.0\right)}^2+\left(\frac{1}{2}\right)M{\left(0\right)}^2=\left(\frac{1}{2}\right)\left(3.0\right){\left(v_{f1}\right)}^2+\left(\frac{1}{2}\right)M{\left(6.0\right)}^2$

We can cancel$\left(\frac{1}{2}\right)$from both sides because it is common on both sides; we get,

$$\begin{gathered} \left(3.0\right){\left(8.0\right)}^2=\left(3.0\right){\left(v_{f1}\right)}^2+M{\left(6.0\right)}^2 \\ 192=\left(3.0\right)){\left(v_{f1}\right)}^2+36M \end{gathered}$$ (2(

Using the value offrom equation (1) in equation (2), we get,

$$\begin{gathered} 192=\left(3.0\right)){\left(8-2M\right)}^2+36M \\ 192=\left(3.0\right)\left(64-32M+4M^2\right)+36M \\ 192=192-96M+12M^2+36M \end{gathered}$$

Solving further as,

$$\begin{gathered} 12M^2-60M=0 \\ 12M=60M \\ M=\frac{60}{12} \\ =5.0\mathrm{kg} \end{gathered}$$

Thus, the mass of the object is, M = 5.0 kg .