A dentist is working on a patient in a reclining chair. The dentist moves at 0.5 m/s towards the patient, and the patient reclines backwards at 0.2 m/s. What is the dentist's speed relative to the pat

Let's consider the direction from the patient towards the dentist as the positive direction.

• Velocity of the dentist ($v_D$): The dentist moves towards the patient. This means the dentist is moving in the negative direction relative to our chosen positive direction. $$v_D = -0.5 \text{ ms}^{-1}$$
• Velocity of the patient ($v_P$): The patient reclines backwards. This means the patient is moving away from the dentist, which is also in the negative direction. $$v_P = -0.2 \text{ ms}^{-1}$$

The relative velocity of object A with respect to object B is given by $v_{AB} = v_A - v_B$.
Here, we want the velocity of the dentist relative to the patient ($v_{DP}$).

$$v_{DP} = v_D - v_P$$

Substitute the values:

$$v_{DP} = (-0.5 \text{ ms}^{-1}) - (-0.2 \text{ ms}^{-1})$$ $$v_{DP} = -0.5 \text{ ms}^{-1} + 0.2 \text{ ms}^{-1}$$ $$v_{DP} = -0.3 \text{ ms}^{-1}$$

Speed is the magnitude of velocity.

$$\text{Speed} = |v_{DP}| = |-0.3 \text{ ms}^{-1}|$$ $$\text{Speed} = 0.3 \text{ ms}^{-1}$$

The dentist's speed relative to the patient is $\text{0.3 ms}^{-1}$.