Motion in Two or Three Dimensions

These flashcards cover key vocabulary related to motion in two or three dimensions, focusing on definitions and concepts that are essential for understanding the subject.

Position Vector

A vector that represents the position of a particle in three-dimensional space, defined by coordinates (x, y, z).

Velocity Vector

A vector that indicates the rate of change of position; it has both a magnitude and a direction.

Instantaneous Velocity

The velocity of an object at a specific instant in time; it is the tangent to the path of the object at that point.

Acceleration Vector

A vector that represents the rate of change of velocity; it can change in both magnitude and direction.

Projectile Motion

The motion of an object that is thrown into the air and is subject to gravitational force, resulting in a curved path.

Trajectory

The path that a projectile follows through space as a function of time.

Relative Velocity

The velocity of an object as observed from a particular reference point, which may be in motion itself.

Uniform Circular Motion

Motion in a circular path at a constant speed, where the acceleration is directed towards the center of the circle.

Instantaneous Acceleration

The acceleration of an object at a specific moment in time, indicating how quickly the velocity changes at that instant.

Average Acceleration

The change in velocity divided by the time over which the change occurs, resulting in a vector quantity.

Position Vector

A vector that represents the position of a particle in three-dimensional space, defined by coordinates (x, y, z).

Velocity Vector

A vector that indicates the rate of change of position; it has both a magnitude and a direction.

Instantaneous Velocity

The velocity of an object at a specific instant in time; it is the tangent to the path of the object at that point.

Acceleration Vector

A vector that represents the rate of change of velocity; it can change in both magnitude and direction.

Projectile Motion

The motion of an object that is thrown into the air and is subject to gravitational force, resulting in a curved path.

Trajectory

The path that a projectile follows through space as a function of time.

Relative Velocity

The velocity of an object as observed from a particular reference point, which may be in motion itself.

Uniform Circular Motion

Motion in a circular path at a constant speed, where the acceleration is directed towards the center of the circle.

Instantaneous Acceleration

The acceleration of an object at a specific moment in time, indicating how quickly the velocity changes at that instant.

Average Acceleration

The change in velocity divided by the time over which the change occurs, resulting in a vector quantity.

Practice Problem 1: Velocity from Position Vector

If the position vector of a particle is given by \vec{r}(t) = (3t^2)\hat{i} + (2t - 1)\hat{j}, what is its velocity vector at t = 2 s?

To find the velocity vector, differentiate the position vector with respect to time:
\vec{v}(t) = \frac{d\vec{r}}{dt} = \frac{d}{dt}(3t^2)\hat{i} + \frac{d}{dt}(2t - 1)\hat{j}
\vec{v}(t) = (6t)\hat{i} + (2)\hat{j}
Now, substitute t = 2 s:
\vec{v}(2) = (6 \cdot 2)\hat{i} + (2)\hat{j} = (12)\hat{i} + (2)\hat{j} m/s.

Practice Problem 2: Projectile Motion (Maximum Height)

A projectile is launched from the ground with an initial speed of 20 m/s at an angle of 30^\circ above the horizontal. What is the maximum height it reaches? (Assume g = 9.8 m/s^2).

First, find the initial vertical velocity:
v{0y} = v0 \sin\theta = 20 \sin(30^\circ) = 20 \cdot 0.5 = 10 m/s.
At maximum height, the vertical velocity vy = 0. Using the kinematic equation vy^2 = v{0y}^2 + 2ay: 0^2 = (10)^2 + 2(-9.8)y{\text{max}}
0 = 100 - 19.6 y{\text{max}} 19.6 y{\text{max}} = 100
y_{\text{max}} = \frac{100}{19.6} \approx 5.10 m.

Practice Problem 3: Relative Velocity

A boat can travel at 10 km/h in still water. If the boat wants to cross a river 2 km wide that has a current of 4 km/h downstream, and the boat aims directly across the river, what is its resultant speed relative to the ground?

The velocity of the boat relative to water is \vec{v}{BW} = 10 km/h (across). The velocity of the water relative to the ground is \vec{v}{WG} = 4 km/h (downstream).
The velocity of the boat relative to the ground is \vec{v}{BG} = \vec{v}{BW} + \vec{v}{WG}. Since the velocities are perpendicular, the magnitude of the resultant speed is: |\vec{v}{BG}| = \sqrt{v{BW}^2 + v{WG}^2} = \sqrt{10^2 + 4^2} = \sqrt{100 + 16} = \sqrt{116} \approx 10.77 km/h.

Practice Problem 4: Uniform Circular Motion (Centripetal Acceleration)

A car is traveling around a circular track with a radius of 50 m at a constant speed of 15 m/s. What is the magnitude of its centripetal acceleration?

For uniform circular motion, the centripetal acceleration is given by the formula:
ac = \frac{v^2}{r} Where v is the speed and r is the radius. ac = \frac{(15 \text{ m/s})^2}{50 \text{ m}} = \frac{225 \text{ m}^2/\text{s}^2}{50 \text{ m}} = 4.5 m/s^2.