Unit 4: Motion in Two and Three Dimensions
Projectiles, circles, and moving reference frames — what changes once motion stops being a straight line.
6 topics · 20 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Position and Displacement Vectors in 2D
- How it worksHow does a position vector's tip trace out a curved path as time moves forward?
- Worked exampleFind the displacement vector for a particle moving from (2 m, 1 m) to (5 m, 6 m).
- Going furtherWhat happens to a displacement vector's direction if the trip takes a longer, curved route to the same endpoint?
Your turn1 of 2
A particle moves from (3 m, 3 m) to (15 m, 10 m). What is the magnitude of its displacement, in meters?Velocity and Acceleration Vectors in 2D
- How it worksHow do you get the velocity vector by taking the derivative of each component of the position vector?
- Worked exampleFind the velocity vector at t = 3 s for r(t) = (2t2) i-hat + (5t) j-hat.
- Common mistakeWhy isn't accelerating the same thing as speeding up once an object's path starts to curve?
Your turn1 of 2
A particle's position is r(t) = (3 t2) i-hat + (6 t) j-hat, with distances in meters and t in seconds. What is the magnitude of its velocity at t = 3 s?Projectile Motion
- How it worksHow do the horizontal and vertical parts of projectile motion stay completely independent of each other?
- Worked exampleFind the range of a projectile launched at 20 m/s at 30 degrees above the horizontal.
- Worked exampleFind the maximum height reached by a ball thrown straight up at 15 m/s.
- Common mistakeWhy doesn't a projectile's horizontal velocity change at all during its flight, ignoring air resistance?
- Going furtherWhat happens to a projectile's range when the launch angle changes from 30 to 60 degrees?
Your turn1 of 2
A projectile is launched at 24 m/s at 54 degrees above the horizontal. Taking g = 9.8 m/s2 and neglecting air resistance, what is its range, in meters?Uniform Circular Motion
- Common mistakeWhy isn't the acceleration in uniform circular motion pointing along the direction of motion?
- How it worksHow does centripetal acceleration always point toward the center even though the object's speed never changes?
- Worked exampleFind the centripetal acceleration of a car rounding a curve of radius 50 m at 20 m/s.
Your turn1 of 2
A car rounds a curve of radius 62 m at a constant 19 m/s. What is its centripetal acceleration, in meters per second squared?Relative Motion
- How it worksHow do you add or subtract velocity vectors to switch an observation between two moving reference frames?
- Worked exampleFind a swimmer's velocity relative to the shore if she swims at 1.5 m/s straight across a river flowing at 2 m/s.
- Going furtherWhat happens to ordinary relative-velocity addition once speeds approach a meaningful fraction of light speed?
Your turn1 of 2
A swimmer swims at 2.24 m/s straight across a river that flows at 2.77 m/s. What is her speed relative to the shore, in meters per second?Tangential and Radial Acceleration
- How it worksHow do you split an object's acceleration into a radial part and a tangential part on a curved path?
- Worked exampleFind the total acceleration of a car on a 100 m radius curve, moving at 15 m/s and speeding up at 2 m/s2.
- Ties togetherHow does splitting acceleration into radial and tangential parts collapse back to ordinary 1D acceleration on a straight road?
Your turn1 of 2
A car moves along a curve of radius 120 m at 20 m/s while speeding up at a tangential rate of 4 m/s2. What is the magnitude of its total acceleration, in meters per second squared?