Answer :
To prove that the momentum transferred from one sphere to the other during a direct impact is given by
[tex]\[ \frac{m m^{\prime}}{m+m^{\prime}}(1+e) \cdot v \][/tex]
where [tex]\( v \)[/tex] is the relative velocity before impact, let's work through the problem step-by-step.
### Step 1: Understand the Given Information
- We have two spheres with masses [tex]\( m \)[/tex] and [tex]\( m' \)[/tex].
- The coefficient of restitution is [tex]\( e \)[/tex].
- The relative velocity before impact is [tex]\( v \)[/tex].
### Step 2: Define Momentum and Coefficient of Restitution
- The coefficient of restitution [tex]\( e \)[/tex] is defined as the ratio of the relative speed after collision to the relative speed before collision along the line of impact. Mathematically,
[tex]\[ e = \frac{\text{relative velocity after impact}}{\text{relative velocity before impact}}. \][/tex]
### Step 3: Calculate Initial and Final Momenta
- Let the initial velocities of the masses [tex]\( m \)[/tex] and [tex]\( m' \)[/tex] be [tex]\( u \)[/tex] and [tex]\( u' \)[/tex], respectively.
- After the impact, let the velocities be [tex]\( v \)[/tex] and [tex]\( v' \)[/tex], respectively.
### Step 4: Use the Conservation of Momentum
The principle of conservation of momentum states that the total momentum before the impact is equal to the total momentum after the impact. So we have:
[tex]\[ m u + m' u' = m v + m' v'. \][/tex]
### Step 5: Use the Coefficient of Restitution Formula
The coefficient of restitution [tex]\( e \)[/tex] can be formulated as:
[tex]\[ e = \frac{v' - v}{u - u'}. \][/tex]
### Step 6: Solve these Equations
We need to formulate equations for the final velocities in terms of the initial velocities, the masses, and [tex]\( e \)[/tex].
From the coefficient of restitution equation:
[tex]\[ v' - v = e (u - u'). \][/tex]
From conservation of momentum:
[tex]\[ m u + m' u' = m v + m' v'. \][/tex]
Let’s solve for [tex]\( v \)[/tex] and [tex]\( v' \)[/tex] using these two equations.
First, rearrange the restitution equation:
[tex]\[ v' = v + e (u - u'). \][/tex]
Substitute [tex]\( v' \)[/tex] from the last equation into the momentum equation:
[tex]\[ m u + m' u' = m v + m' (v + e (u - u')). \][/tex]
Expanding and rearranging:
[tex]\[ m u + m' u' = m v + m' v + m' e (u - u'). \][/tex]
[tex]\[ m u + m' u' = (m + m') v + m' e u - m' e u'. \][/tex]
Collecting the terms involving [tex]\( v \)[/tex]:
[tex]\[ (m + m') v = m u + m' u' - m' e u + m' e u'. \][/tex]
[tex]\[ (m + m') v = m u + m' u' (1 + e) - m' e u. \][/tex]
Thus,
[tex]\[ v = \frac{m u + m' u' (1 + e) - m' e u}{m + m'}. \][/tex]
To find the momentum transferred from one sphere to another, we simply need to multiply this relative velocity by the reduced mass (effective mass in a system with two masses):
The reduced mass [tex]\( \mu \)[/tex] is given by:
[tex]\[ \mu = \frac{m m'}{m + m'}. \][/tex]
Thus, the momentum transferred:
[tex]\[ \text{Momentum Transferred} = \mu \cdot (1 + e) \cdot \text{relative velocity before impact} = \frac{m m'}{m + m'} (1 + e) \cdot v. \][/tex]
Therefore, we have proved that the momentum transferred from one sphere to the other during a direct impact is:
[tex]\[ \frac{m m'}{m + m'} (1 + e) \cdot v. \][/tex]
[tex]\[ \frac{m m^{\prime}}{m+m^{\prime}}(1+e) \cdot v \][/tex]
where [tex]\( v \)[/tex] is the relative velocity before impact, let's work through the problem step-by-step.
### Step 1: Understand the Given Information
- We have two spheres with masses [tex]\( m \)[/tex] and [tex]\( m' \)[/tex].
- The coefficient of restitution is [tex]\( e \)[/tex].
- The relative velocity before impact is [tex]\( v \)[/tex].
### Step 2: Define Momentum and Coefficient of Restitution
- The coefficient of restitution [tex]\( e \)[/tex] is defined as the ratio of the relative speed after collision to the relative speed before collision along the line of impact. Mathematically,
[tex]\[ e = \frac{\text{relative velocity after impact}}{\text{relative velocity before impact}}. \][/tex]
### Step 3: Calculate Initial and Final Momenta
- Let the initial velocities of the masses [tex]\( m \)[/tex] and [tex]\( m' \)[/tex] be [tex]\( u \)[/tex] and [tex]\( u' \)[/tex], respectively.
- After the impact, let the velocities be [tex]\( v \)[/tex] and [tex]\( v' \)[/tex], respectively.
### Step 4: Use the Conservation of Momentum
The principle of conservation of momentum states that the total momentum before the impact is equal to the total momentum after the impact. So we have:
[tex]\[ m u + m' u' = m v + m' v'. \][/tex]
### Step 5: Use the Coefficient of Restitution Formula
The coefficient of restitution [tex]\( e \)[/tex] can be formulated as:
[tex]\[ e = \frac{v' - v}{u - u'}. \][/tex]
### Step 6: Solve these Equations
We need to formulate equations for the final velocities in terms of the initial velocities, the masses, and [tex]\( e \)[/tex].
From the coefficient of restitution equation:
[tex]\[ v' - v = e (u - u'). \][/tex]
From conservation of momentum:
[tex]\[ m u + m' u' = m v + m' v'. \][/tex]
Let’s solve for [tex]\( v \)[/tex] and [tex]\( v' \)[/tex] using these two equations.
First, rearrange the restitution equation:
[tex]\[ v' = v + e (u - u'). \][/tex]
Substitute [tex]\( v' \)[/tex] from the last equation into the momentum equation:
[tex]\[ m u + m' u' = m v + m' (v + e (u - u')). \][/tex]
Expanding and rearranging:
[tex]\[ m u + m' u' = m v + m' v + m' e (u - u'). \][/tex]
[tex]\[ m u + m' u' = (m + m') v + m' e u - m' e u'. \][/tex]
Collecting the terms involving [tex]\( v \)[/tex]:
[tex]\[ (m + m') v = m u + m' u' - m' e u + m' e u'. \][/tex]
[tex]\[ (m + m') v = m u + m' u' (1 + e) - m' e u. \][/tex]
Thus,
[tex]\[ v = \frac{m u + m' u' (1 + e) - m' e u}{m + m'}. \][/tex]
To find the momentum transferred from one sphere to another, we simply need to multiply this relative velocity by the reduced mass (effective mass in a system with two masses):
The reduced mass [tex]\( \mu \)[/tex] is given by:
[tex]\[ \mu = \frac{m m'}{m + m'}. \][/tex]
Thus, the momentum transferred:
[tex]\[ \text{Momentum Transferred} = \mu \cdot (1 + e) \cdot \text{relative velocity before impact} = \frac{m m'}{m + m'} (1 + e) \cdot v. \][/tex]
Therefore, we have proved that the momentum transferred from one sphere to the other during a direct impact is:
[tex]\[ \frac{m m'}{m + m'} (1 + e) \cdot v. \][/tex]
