Two-Dimensional Kinematics

A set of practice questions and answers covering key concepts from two-dimensional kinematics: vector representations, graphical and analytical addition, independence of motion, projectile motion, and relative velocity.

What are the two components of motion in two-dimensional kinematics?

Horizontal and vertical components.

What does the independence of horizontal and vertical motions mean in two-dimensional motion?

The horizontal and vertical components are independent; motion in one does not affect the other (gravity acts vertically; neglect air resistance for horizontal motion changes).

How is a vector represented graphically in two dimensions?

As an arrow with length proportional to its magnitude and pointed in the vector’s direction.

What is displacement and how is it decomposed in two dimensions?

Displacement is the straight-line vector from initial to final position; its components are the horizontal (x) and vertical (y) components.

Explain the head-to-tail method for adding vectors.

Place the tail of each successive vector at the head of the previous one; the resultant is drawn from the tail of the first to the head of the last.

Are vector additions commutative?

Yes. The resultant does not depend on the order in which vectors are added.

How is vector subtraction performed graphically?

By adding the negative of the vector to be subtracted.

What happens when you multiply a vector by a scalar?

The magnitude scales by the absolute value of the scalar; the direction remains the same if the scalar is positive and reverses if negative.

What does resolving a vector into components mean?

Expressing the vector as the sum of perpendicular components along chosen axes (e.g., x and y).

If a vector has magnitude v and direction theta from the x-axis, what are its components?

vx = v cos theta; vy = v sin theta.

What is projectile motion?

Motion of an object acted on only by gravity (negligible air resistance), analyzed as horizontal and vertical motions.

In projectile motion, what is the horizontal acceleration?

Zero (neglecting air resistance).

What are the vertical-motion equations for projectile motion with constant gravity?

y = v0y t - (1/2) g t^2; v_y = v0y - g t.

How do you compute the range of a projectile on level ground neglecting air resistance?

R = v^2 sin(2 theta) / g, assuming launch and landing at the same height.

For a fixed initial speed, which two launch angles give the same range on level ground?

Theta and 90 degrees minus theta; their sum is 90 degrees.

What angle maximizes the range of a projectile on level ground (neglecting air resistance)?

45 degrees.

What is relative velocity?

The velocity of an object as observed from a particular reference frame; the total velocity is the sum of velocity relative to the medium and the medium’s velocity relative to the observer.

In a river crossing problem, what is the boat’s total velocity relative to the shore?

The vector sum of the boat’s velocity relative to the water and the river’s velocity relative to the shore.

How do you compute velocity components when given magnitude and direction?

vx = v cos theta; vy = v sin theta.

How do you solve wind-velocity problems?

Treat total velocity as the sum of the plane’s velocity relative to the air and the wind; resolve into components to find the wind’s magnitude and direction.

What is the apex in projectile motion, and how is maximum height related to initial vertical velocity?

The apex is the highest point; maximum height H = v0y^2 /(2g) for zero initial height.

What is the projectile range formula and its limits due to Earth’s curvature?

R = v^2 sin(2 theta)/g for level ground neglecting air resistance; for long ranges, Earth curvature and gravity variation modify the result.

What is the trajectory in projectile motion?

The path of a projectile through the air.

What is a vector?

A quantity that has both magnitude and direction.

What is a scalar?

A quantity that has magnitude only.

What is the resultant in vector addition?

The vector sum of two or more vectors.

How do you find the magnitude and direction of a resultant from its components?

Magnitude: R = sqrt(Rx^2 + Ry^2); direction: theta = arctan(Ry / Rx) relative to the x-axis.

What does the term 'component vectors' mean?

The projections of a vector along the x- and y-axes (e.g., vx and vy).