Answer:
a) 0.62 m/s
b) No
c) 0.27 joules
Explanation:
a) Total linear momentum (P) is the sum of the individual momenta of Cart 1 and Cart 2:
[tex]\overrightarrow{p}=\overrightarrow{p_{1}}+\overrightarrow{p_{2}} [/tex]
if it's zero
[tex]\overrightarrow{p_{1}}+\overrightarrow{p_{2}}=0 [/tex]
Momentum is the product of mass and velocity so:
[tex]m_{1}\overrightarrow{v_{1}}+m_{2}\overrightarrow{v_{2}} = 0[/tex]
because the direction of the Carts is opposite to each other, we should use a negative sing for one of the velocities (no matter which)
[tex]m_{1}v_{1}+m_{2}(-v_{2}) =0[/tex]
[tex]m_{1}v_{1}-m_{2}v_{2}=0 [/tex]
solving for [tex]v_{2} [/tex]
[tex]m_{1}v_{1}=m_{2}v_{2} [/tex]
[tex]v_{2}=\frac{m_{1}v_{1}}{m_{2}}=\frac{(0.47)(0.8)}{0.61} [/tex]
[tex]v_{2}=0.62\frac{m}{s} [/tex]
b) No, kinetic energy and linear momentum are different things, one of the most important differences is that kinetic energy is a scalar quantity and linear momentum is a vector quantity.
c)System kinetic energy is the sum of the individual kinetic energies
[tex]K=K_{1}+K{2}=\frac{m_{1}v_{1}^{2}}{2}+\frac{m_{2}v_{2}^{2}}{2} [/tex]
[tex]K= \frac{(0.47)(0.8)^{2}}{2}+\frac{(0.61)(0.62)^{2}}{2}=0.27J[/tex]
It's different of zero!
a. Since the total momentum of the system is to be zero, the initial speed of Cart 2 is equal to 0.62 m/s.
b. No, the kinetic energy of the system is not equal to zero because energy can not be destroyed according to the law of conservation of energy.
c. The system's kinetic energy is equal to 0.268 Joules.
Given the following data:
a. To determine the initial speed of Cart 2, if total momentum of the system is to be zero:
Since the total momentum of the system is zero, the momentum of the two carts is given by the formula:
[tex]M_1V_1 =M_2V_2[/tex]
Where:
Making [tex]V_2[/tex] the subject of formula, we have:
[tex]V_2 = \frac{M_1V_1 }{M_2}[/tex]
Substituting the given parameters into the formula, we have;
[tex]V_2 = \frac{0.47 \times 0.8 }{0.61}\\\\V_2 = \frac{0.376 }{0.61}\\\\V_2 =0.62 \;m/s[/tex]
b. No, the kinetic energy of the system is not equal to zero because energy can not be destroyed according to the law of conservation of energy.
c. To determine the system's kinetic energy:
[tex]T.E = K.E_i + K.E_f\\\\T.E = \frac{1}{2} M_1V_1^2 + \frac{1}{2} M_2V_2^2\\\\T.E = \frac{1}{2} \times 0.47 \times (0.8)^2 + \frac{1}{2} \times 0.61 \times (0.62)^2\\\\T.E = 0.235 \times 0.64 + 0.305 \times 0.3844\\\\T.E = 0.1504 + 0.1172[/tex]
System's kinetic energy = 0.268 Joules.
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