A child boards a bus for summer camp. His mom is waiting next to the bus. The bus begins moving 3 m/s east with respect to the ground. The child is walking 0.5 m/s west towards the back of the bus with respect to the bus. The mother begins running 1 m/s east with respect to the ground, chasing after the bus and thoroughly humiliating her child. What is the child’s velocity a) relative to the ground? b) relative to the mother?
The correct answer and explanation is:
Part a) Child’s Velocity Relative to the Ground
To find the child’s velocity relative to the ground, we need to consider both the motion of the bus and the child’s motion within the bus.
The bus is moving east with a velocity of 3 m/s relative to the ground. The child, however, is walking towards the back of the bus (westward) at 0.5 m/s relative to the bus. Since the bus is moving east and the child is moving west within the bus, we subtract the child’s velocity from the bus’s velocity to find the child’s velocity relative to the ground.
- Velocity of the bus relative to the ground = 3 m/s east
- Velocity of the child relative to the bus = 0.5 m/s west (which is negative relative to the bus’s direction)
Thus, the child’s velocity relative to the ground is: Child’s velocity relative to the ground=3 m/s−0.5 m/s=2.5 m/s east\text{Child’s velocity relative to the ground} = 3 \, \text{m/s} – 0.5 \, \text{m/s} = 2.5 \, \text{m/s east}
Part b) Child’s Velocity Relative to the Mother
Next, we calculate the child’s velocity relative to the mother, who is running east at 1 m/s relative to the ground. The child’s velocity relative to the ground is 2.5 m/s east, as calculated earlier, and the mother’s velocity relative to the ground is 1 m/s east.
To find the relative velocity between the child and the mother, we subtract the mother’s velocity from the child’s velocity relative to the ground: Child’s velocity relative to the mother=2.5 m/s east−1 m/s east=1.5 m/s east\text{Child’s velocity relative to the mother} = 2.5 \, \text{m/s east} – 1 \, \text{m/s east} = 1.5 \, \text{m/s east}
Conclusion
a) The child’s velocity relative to the ground is 2.5 m/s east.
b) The child’s velocity relative to the mother is 1.5 m/s east.
In this situation, the mother is chasing after the bus and, despite her efforts, the child is still moving east at a faster rate than her.