Question

A 0.060 kg mass is used to accelerate a cart. At t = 1.10 ± 0.01 s, the 0.50 kg cart is located at x = 0.2 m and is at rest (that is, v0 = 0 m/s). At t = 2.30 ± 0.01 s, the cart is located at 1.2 m, traveling at vf = 1.5 ± 0.2 m/s. Following the procedure in the first part of the lab, calculate the impulse and the change in the system’s momentum. Are they within the uncertainty of each other? 2. If a cart of mass m1 = 0.60 kg, traveling at v10 = 0.45 ± 0.03 m/s, collides elastically with a cart of mass m2 = 0.30 kg initially at rest, what is the final velocity of the first cart, and its uncertainty? 3. If the two carts from Question 2 stick together after the collision, what is the final velocity of the first cart, and its uncertainty

Answer 1

The definitions of momentum and impluse and their conservation allow finding the results for the different questions are:

1)  The two quantities are in the same range

2) Elastic shock v =( 0.15 ±0.01 )  m/s

3) inelastic shock  v = (0.30 ± 0.02) m / s

1) Momentum is a magnitude that indicates the change in the momentum of an object

           I = ∫ F . dt = Δp

Where I is the momentum, F the force, t the time, Δp the change in momentum

Let's look for the average momentum

          I = F Δt

          I = m g Δt

          I = 0.060 9.8 (2.30 - 1.1)

          I = 0.71 kg m /s

The uncertainty is the statistical fluctuation of the magnitude, it is given by

          ΔI = $\frac{dI}{dt}$  

          ΔI = mg (Δt + Δt₀)

          ΔI = 0.06 9.8 (0.02)

          ΔI = 0.01

The result for momentum is

           I = (0.71 ± 0.01) kg m / s

They ask to compare with the variation momentum

         Δp = m v - m v₀

         

Let's calculate

          Δp = 0.5 (1.5 -0)

          Δp = 0.75 kg m / s

We calculate the uncertainty

          Δp = $\frac{dp}{dv}$

          Δp = m (Δv + Δv)

          Δp = 0.5 (0.2 + 0)

          Δp = 0.1 kg m/s

The result for the variation momentum  is

           ΔP = (0.8 ± 0.2 ) kg m / s

We can see that the fluctuation of the momentum change is high, therefore it is within the fluctuation range of the impulse, the two quantities are in the same range

2) In this case we define a system formed by the two cars, for this system the momentum is conserved since the forces during the collision are internal

Initial instant. Before the crash

          $p_o = m_1 v_{1o}$  

Final moment. After the crash

         $pf = m_1 v_{1f} + m_2v_{2f}$

The momentun is preserved

          $p_o = p_f\\m_1 v_{1o} = m_1 v_{1f} + m_2 v_{2f}$

As they indicate that the shock is elastic, the kinetic energy is conserved

Initial instant

           K₀ = ½ m₁ $v_{1o}^2$  

Final moment

            $K_f$ =  ½ m₁ $v_{1f}^2$ + ½ m₂ $v_{2f}^2$

Energy is conserved

           $K_o = K_f\\m_1 v_{1o}^2 = m_1 v_{1f}^2 + m_2 v_{2f}^2$

           

We write the system of equations

       $\frac{m_1}{m_2} (v_{1o} - v_{1f} ) = v_{2f}\\\frac{m_1}{m_2} (v_{1o}^2 - v_{1f}^2 ) = v_{2f}^2$

       

We solve the system of equations

        ( $( v({1o}^2 - v_{1f}^2 ) = \frac{m_1}{m_2} \ ( v_{1o} - v(1f} )^2$

         $( v_{1o} + v_{1f} ) = \frac{m_1}{m_2} (v_{1o} - v_{1f} )\\v_{1f} ( 1 + \frac{m_1}{m_2} ) = v_{1o} ( \frac{m1}{m_2} - 1)$

We substitute

          $v_{1f} ( 1 + 2) =0.45 (2-1)\\v_{1f} = 0.15 m/s$

We look for uncertainty

         Δv = $\frac{d v_{1f}}{dv_{1o}}$

         Δv = $\frac{1}{3} \Delta v_{1o}$

         Δv = 0.01 m/s    

           

The result is

         v =( 0.15 ±0.01 )  m/s

3) In this case the two vehicles remain united after the collision, inelastic shock

Let's use conservation of the moment

          p₀ = p_f

          m₁ v₁₀ = (m₁ + m₂) v

          v = $\frac{m_1}{m_1+m_2}$   v₁₀

 

We calculate

          v = $\frac{0.6}{0.9 } \ 0.45$  

          v = 0.3 m / s

we look for uncertainty

         Δv = $\frac{d v}{d v_{1o}}$

         Δv = $\frac{m_1}{m_1+m_2} \ \Delta v_{1o}$

           

         ΔV = $\frac{0.6 }{0.9} \ 0.03$

         Δv = 0.02 m / s

The result is

         v = (0.30 ± 0.02) m / s

In conclusion with the definition of momentum and impulse and its conservation we can find the results for the different questions are:

      1)  The two quantities are in the same range

      2) Elastic shock v =( 0.15 ±0.01 )  m/s

      3) inelastic shock  v = (0.30 ± 0.02) m / s

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