Exercise \(\PageIndex{1}\): Determine velocity in another frame
Two objects approach each other as shown in the animation. Before they collide, what is the speed of the green object as measured in the reference frame of the red object (position is given in meters and time is given in seconds)? Restart.
Exercise \(\PageIndex{2}\): You are rowing a boat across a river
You are rowing a boat across a river (position is given in meters and time is given in seconds). The river current flows from left to right in the animation. Your goal is to reach your favorite mooring spot on the opposite river bank (shown in red). By rowing, you control the (constant, vertical) velocity of the boat with respect to the water. Restart.
- What is the (horizontal) velocity of the water with respect to the banks?
- What is the magnitude of the velocity of the boat with respect to the banks?
- What angle, \(\theta\), does the boat's velocity vector make with the lower bank?
Problem authored by Steve Mellema and Chuck Niederriter.
Exercise \(\PageIndex{3}\): Which is an inertial frame?
Two space aliens measure the position of an orange star from different space ships. Alien 1 records data as \(x_{1}\) and alien 2 records data as \(x_{2}\) (position is given in meters and time is given in seconds). You are viewing the same star from an inertial reference frame. Assume that you and the aliens have agreed to use the same distance and time units. You suspect that one of your alien friends is in a noninertial reference frame. Restart.
- Which alien is in an inertial reference frame and which alien is in a noninertial frame?
- Find the Galilean transformation from your frame into the inertial reference frame.
Exercise \(\PageIndex{4}\): Determine relative velocities
Two physics coworkers measure the position of the object shown from different inertial reference frames (position is given in meters and time is given in seconds). Coworker 1 records data as \(x_{1}\) and \(v_{1}\) and coworker 2 records data as \(x_{2}\) and \(v_{2}\). Assume both coworkers have agreed to use meters and seconds to measure distance and time. Restart.
- What is the relative speed of coworker 1 with respect to coworker 2?
- Find the Galilean transformation that transforms the measurements of coworker 1 into those of coworker 2.
Exercise \(\PageIndex{5}\): Two carts connected by a spring
Two physics coworkers measure the position of the object shown from different inertial reference frames (position is given in meters and time is given in seconds). Coworker 1 records data as \(x_{1}\) and \(v_{1}\) and coworker 2 records data as \(x_{2}\) and \(v_{2}\). Assume both coworkers have agreed to use meters and seconds to measure distance and time. Restart.
- What is the relative speed of coworker 1 with respect to coworker 2?
- Find the Galilean transformation that transforms the measurements of coworker 1 into those of coworker 2.
Exercise \(\PageIndex{6}\): Determine the equation describing relative motion of a cart
Two carts start at rest on similar air tracks as shown in the animation (position is given in meters and time is given in seconds). Restart. Write an equation for the x position of the green cart as seen from the orange cart.
Exercise \(\PageIndex{7}\): Determine the relative velocities of the boats
Assume you are sitting on the shore of a lake and observe two boats (not shown to scale) drifting in the lake. Another student is riding in one of the boats (position is given in meters and time is given in seconds). Restart.
- For each of the animations, find the velocity of each boat as seen from the shore.
- Determine the velocity of the red boat as seen by the student riding in the green boat for each of the animations. This is the relative velocity between the two boats.
You can simulate running along the shore by typing a velocity, \(-15\text{ m/s} < v < 15\text{ m/s}\), into the input field before you select an animation.
- Does changing the reference frame, i.e., running along the shore, change the relative velocities of the two boats? Try reference frame velocities of \(+2\text{ m/s}\) and \(-2\text{ m/s}\).
- For each of the animations, what velocity should you enter so that the red boat appears stationary?
- For each of the animations, what velocity should you enter so that the green boat appears stationary?
Exercise \(\PageIndex{8}\): Determine the transformation between frames of reference
A physics student measures the position and velocity of the ball shown in the animation and reports the results to you in the table shown. You are viewing the same experiment from another reference frame (position is given in meters and time is given in seconds). Your values are shown next to the ball in the animation. Assume that your reference frame is an inertial reference frame and that both you and the other student use the same units for time and distance. Restart.
Do the following for each of the animations:
- Determine if the other student is in an inertial reference frame.
- Determine the transformation that transforms the \(x\) and \(v\) data from your frame to her frame.
Exercise \(\PageIndex{9}\): Two airplanes; one experiences a head/tail wind
Two airplanes (not shown to scale) travel the same round-trip distance between two cities (time is given in hours). Both airplanes have the same air speed, but one airplane (the top airplane with the blue wingtip) travels faster or slower relative to the ground because it is subject to a headwind and a tailwind. A positive wind velocity means a tailwind on the outbound part of the trip and a headwind on the inbound part of the trip. What is the ratio of the wind speed to the air speed for the top airplane? Restart.
Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.
