A 30 g dart traveling horizontally hits and sticks in the back of a 500 g toy car, causing the car to roll forward at 1.4 m/s. What was the speed of the dart?
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1
Identify the masses and velocities involved. Let the mass of the dart be $m_d = 30 \text{ g}$ (convert this to kilograms for calculations), the mass of the toy car be $m_c = 500 \text{ g}$ (also convert to kilograms), the initial velocity of the car $v_{c,i} = 0 \text{ m/s}$ (since it's initially at rest), and the final combined velocity $v_f = 1.4 \text{ m/s}$.
Apply the principle of conservation of momentum. The total momentum before the collision must equal the total momentum after the collision. Set up the equation: $m_d \cdot v_{d,i} + m_c \cdot v_{c,i} = (m_d + m_c) \cdot v_f$.
Substitute the known values into the momentum conservation equation, remembering to convert the masses from grams to kilograms.
Solve the equation for the initial velocity of the dart $v_{d,i}$. Rearrange the equation to isolate $v_{d,i}$ on one side: $v_{d,i} = \frac{(m_d + m_c) \cdot v_f - m_c \cdot v_{c,i}}{m_d}$.
Calculate the value of $v_{d,i}$ using the values substituted into the equation. This will give you the speed of the dart before the collision.