Exercises: Reference Frames and Relative Motion

A train moves east at $20\ \text{m/s}$ relative to the ground. Mia walks toward the front of the train at $1.5\ \text{m/s}$ relative to the train. Find Mia's velocity relative to the ground.

On a straight highway, car A moves east at $25\ \text{m/s}$ and car B moves east at $30\ \text{m/s}$, both measured relative to the road. Find the velocity of car B relative to car A, and state what a passenger in car A would observe about car B.

Liam is standing still on a platform. A runner passes him at $4.0\ \text{m/s}$ to the right. Describe the velocity of Liam relative to the runner (magnitude and direction), and explain why the result is not the same as the velocity of the runner relative to Liam.

A boat can travel at $5.0\ \text{m/s}$ in still water. It heads straight across a river $120\ \text{m}$ wide, pointing perpendicular to the bank, while the current flows downstream at $3.0\ \text{m/s}$. (a) Find the speed of the boat relative to the bank. (b) Find the time to cross. (c) Find how far downstream the boat lands.

Two cars approach an intersection. Car 1 travels east at $20\ \text{m/s}$ and car 2 travels north at $15\ \text{m/s}$, both relative to the ground. Find the magnitude and direction (angle measured from east) of the velocity of car 2 relative to car 1.

Sofia is on a moving walkway at an airport that is $60\ \text{m}$ long and moves at $1.2\ \text{m/s}$ relative to the floor. She walks at $0.8\ \text{m/s}$ relative to the walkway in the same direction as the walkway. Ethan walks on the floor beside the walkway at $1.4\ \text{m/s}$ in the same direction. They start together at the beginning of the walkway. (a) Find the time Sofia takes to reach the end. (b) Find how far ahead of or behind Sofia, measured along the floor, Ethan is at that moment.

A swimmer can swim at $1.5\ \text{m/s}$ in still water. She wants to cross a river $90\ \text{m}$ wide and land at the point directly opposite her starting point. The current is $0.90\ \text{m/s}$. (a) Determine the angle upstream from the perpendicular she must aim. (b) Find her speed relative to the bank and the time to cross. (c) If she instead aimed straight across, determine how far downstream she would land and explain why she would reach the far bank sooner than in part (b).

A plane's airspeed is $200\ \text{m/s}$ and it heads due north. A wind blows from the west toward the east at $30\ \text{m/s}$ relative to the ground. (a) Find the plane's velocity relative to the ground (magnitude and direction east of north). (b) The pilot needs to travel due north relative to the ground. Determine the heading the pilot must use, and the resulting ground speed. (c) Compare the times to fly $100\ \text{km}$ due north in the situation of part (b) versus a day with no wind, and explain the difference.