4.65m/s is her velocity as the cart starts moving.
We have given the mass of the girl is [tex]$40 \mathrm{~kg}$[/tex] and the mass of the cart is [tex]$3 \mathrm{~kg}$[/tex].
[tex]$\begin{aligned}&m_{g i r l}=40 \mathrm{~kg} \\&\Rightarrow m_{c a r t}=3 \mathrm{~kg}\end{aligned}$[/tex]
We have given the horizontal velocity of the girl before jumping on the cart is [tex]$5 \mathrm{~m} \cdot \mathrm{s}^{-1}$[/tex].
[tex]$v_{\text {girl }}=5 \mathrm{~m} \cdot \mathrm{s}^{-1}$[/tex]
Initially, the cart is at rest. Hence, the velocity of the cart before the girl jumps on it is [tex]$0 \mathrm{~m} \cdot \mathrm{s}^{-1}$[/tex].
[tex]$v_{\text {cart }}=0 \mathrm{~m} \cdot \mathrm{s}^{-1}$[/tex]
We have asked to determine the velocity of the cart after the girl jumps on it which is the velocityof the girl-cart system as the girl jumps on the cart and the cart starts moving. The collision between the girl and the cart is inelastic.
According to the law of conservation of the linear momentum, the initial linear momentum of the girl and the cart is equal to the linear momentum of the girl-cart system. Hence, the equation (2) for the law of conservation of the linear momentum for inelastic collision becomes
[tex]$m_{\text {girl }} v_{\text {girl }}+m_{\text {cart }} v_{\text {cart }}=\left(m_{\text {girl }}+m_{\text {cart }}\right) V$[/tex]
Substitute [tex]$40 \mathrm{~kg}$[/tex] for [tex]$m_{\text {girl }}, 3 \mathrm{~kg}$[/tex] for [tex]$m_{\text {cart }}, 5 \mathrm{~m} \cdot \mathrm{s}^{-1}$[/tex] for [tex]$v_{\text {girl }}$[/tex] and [tex]$0 \mathrm{~m} \cdot \mathrm{s}^{-1}$[/tex] for [tex]$v_{\text {cart }}$[/tex] in the above equation.
[tex]$\begin{aligned}&(40 \mathrm{~kg})\left(5 \mathrm{~m} \cdot \mathrm{s}^{-1}\right)+(3 \mathrm{~kg})\left(0 \mathrm{~m} \cdot \mathrm{s}^{-1}\right)=(40 \mathrm{~kg}+3 \mathrm{~kg}) V \\&\Rightarrow 200+0=(43) V \\&\Rightarrow V=\frac{200}{43} \\&\therefore V=4.65 \mathrm{~m} \cdot \mathrm{s}^{-1}\end{aligned}$[/tex]
Hence, the velocity of the cart after the girl jumps on it is [tex]4.65 \mathrm{~m} \cdot \mathrm{s}^{-1}$.[/tex]
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