Question

A toy car with a mass of \(120 \mathrm{g}\) moves to the right with a speed of \(0.75 \mathrm{m} / \mathrm{s} .\) A small child drops a \(30.0-\mathrm{g}\) piece of clay onto the car. The clay sticks to the car and the car continues to the right. What is the change in speed of the car? Consider the frictional force between the car and the ground to be negligible.

Short answer

Answer: The change in speed of the toy car is -0.15 m/s.

Step-by-step solution

Identify the initial and final states of the system

Initially, the toy car is moving to the right with a speed of \(0.75 m/s\), and the clay piece is at rest. Finally, the clay sticks to the car, and they both move together to the right with some new speed.

Write down the conservation of momentum equation

The conservation of momentum equation states that the total momentum before an event (in this case, the clay falling on the car) is equal to the total momentum after the event. Mathematically, this can be written as: $$m_1v_1 + m_2v_2 = (m_1 + m_2) v_f$$, where $$m_1, v_1$$ and $$m_2, v_2$$ are the masses and velocities of the car and clay piece, respectively, and $$v_f$$ is the final velocity of the car and clay system.

Calculate the initial momentum of the system

The car has an initial mass of \(0.12 kg\) (converting from grams), and moves with a speed of \(0.75 m/s\). The initial momentum for the car is thus $$m_1 v_1 = (0.12 kg)(0.75 m/s) = 0.09 kg \times m/s$$. Since the clay is at rest initially, its momentum is zero ($$m_2 v_2 = 0$$).

Calculate the final momentum of the system

Since the initial momentum of the system is equal to the final momentum, we know that the final momentum is also equal to \(0.09 kg \thinspace m/s\).

Calculate the final velocity of the system

We can now find the final velocity of the car and clay system. We know $$m_1 + m_2 = 0.12 kg + 0.03 kg = 0.15 kg$$, and the total final momentum is $$0.09 kg \times m/s$$. Divide the final momentum by the total mass to find the final velocity: $$v_f = \frac{0.09 \, kg \thinspace m/s}{0.15 kg} = 0.6 m/s$$.

Determine the change in speed

The change in speed can be calculated by taking the difference between the final and initial speeds of the car: $$\Delta v = v_f - v_1 = 0.6 m/s - 0.75 m/s = -0.15 m/s$$. The negative sign indicates that the speed of the car has decreased by \(0.15 m/s\) after the clay piece has stuck onto it.