U1L1E3: Components of Two-Dimensional Motion

In two-dimensional motion, the horizontal component of motion of a projectile:

• Varies with time due to gravity

• Remains constant if air resistance is negligible

• Is affected by the weight of the object

• Increases as the object goes higher

Answer explanation

In two-dimensional motion, the horizontal component remains constant if air resistance is negligible because there are no horizontal forces acting on the projectile. Gravity only affects the vertical motion.

The vertical component of a projectile's velocity:

• Is zero at all times

• Is constant

• Changes due to acceleration from gravity

• Changes only if the initial velocity is zero

Answer explanation

The vertical component of a projectile's velocity changes due to acceleration from gravity, which acts downward, affecting the velocity throughout the projectile's flight.

Which of the following statements about vector components is true?

• A vector can be split into horizontal and vertical components using trigonometry.

• The vertical component of a vector is always larger than the horizontal component.

• The horizontal component is always equal to the magnitude of the vector.

• Vector components cannot be combined to find the original vector.

Answer explanation

The correct statement is that a vector can be split into horizontal and vertical components using trigonometry. This allows for the analysis of the vector's effects in different directions.

If a vector A makes an angle θ with the horizontal axis, the horizontal component of vector A is:

• A_y=∣A∣sin⁡θ

• A_x=∣A∣cos⁡θ

• A_y=∣A∣tan⁡θ

• A_x=∣A∣cot⁡θ

Answer explanation

The horizontal component of vector A is given by A_x = |A|cos(θ). This is because the cosine function relates the adjacent side (horizontal component) to the hypotenuse (the vector's magnitude) in a right triangle.

In two-dimensional motion, if the initial velocity of a projectile is v₀ and the angle of projection is θ, the initial vertical component of velocity is:

• v₀cos⁡θ

• v₀sin⁡θ

• v₀tan⁡θ

• v₀cot⁡θ

Answer explanation

In projectile motion, the initial vertical component of velocity is calculated using the sine function. Thus, the initial vertical component is v₀sinθ, making this the correct choice.

Which trigonometric function is used to find the vertical component of a vector when the angle and magnitude are known?

• Sine

• Cosine

• Tangent

• Cotangent

Answer explanation

The sine function relates the angle of a right triangle to the ratio of the length of the opposite side (vertical component) to the hypotenuse (magnitude). Thus, to find the vertical component, use the sine of the angle.

The resultant vector of two perpendicular vectors A_x and A_y can be found using:

• Ax+A_y

• tan⁡⁻¹(A_y/A_x)

• A_xsin⁡θ+A_ycos⁡θ

If a projectile is launched at an angle of 45 degrees with an initial velocity of v0, what is the relationship between the horizontal and vertical components of the initial velocity?

• The horizontal component is twice the vertical component.

• The vertical component is twice the horizontal component.

• Both components are equal.

• The vertical component is zero.

Answer explanation

When a projectile is launched at a 45-degree angle, the horizontal and vertical components of the initial velocity are equal. This is because the angle divides the initial velocity equally between the two directions.

What is the shape of the trajectory of a projectile when launched at an angle?

• A straight line

• A circle

• A parabola

• An ellipse

Answer explanation

The trajectory of a projectile launched at an angle follows a parabolic path due to the influence of gravity acting on it. This shape results from the combination of horizontal motion and vertical acceleration.

When a projectile reaches its maximum height, the vertical component of its velocity is:

• Zero

• Maximum

• Equal to the horizontal component

• Constant

Answer explanation

When a projectile reaches its maximum height, the vertical component of its velocity becomes zero because it momentarily stops moving upward before descending. Thus, the correct answer is 'Zero'.