Question

A ball rolls horizontally off a balcony at an initial velocity of 5.00 m/s and lands on the ground 10.0 m below.Find the following answers and give them with 3 sig. figs. A) The time the ball is in the air is _______ seconds. B) The magnitude of the final velocity in the y- direction is _______m/s. C) The magnitude of the total final velocity is _________m/s. D) The angle the ball makes with the horizon is _______ degrees. (Give an answer between 0 and 90 degrees, the positive value)

Answer 1

Answer 2

To solve the given problem, we can use the equations of motion for an object in free fall under constant acceleration due to gravity.

Given: Initial velocity in the horizontal direction (Vx) = 5.00 m/s Vertical displacement (Δy) = -10.0 m (negative because the ball is falling downward) Acceleration due to gravity (g) = 9.81 m/s²

A) The time the ball is in the air can be found using the vertical displacement and the equation:

Δy = Vyi × t + (1/2) × g × t²

Since the initial vertical velocity (Vyi) is zero, the equation simplifies to:

Δy = (1/2) × g × t²

Solving for t:

t² = (2 × Δy) / g

t² = (2 × -10.0 m) / 9.81 m/s²

t² ≈ -2.04

Since time cannot be negative, we discard the negative solution. Therefore:

t ≈ √2.04 ≈ 1.43 seconds (to 3 significant figures)

B) The magnitude of the final velocity in the y-direction (Vy) can be found using the equation:

Vy = Vyi + g × t

Vy = 0 + 9.81 m/s² × 1.43 s

Vy ≈ 14.0 m/s (to 3 significant figures)

C) The magnitude of the total final velocity can be found using the Pythagorean theorem:

V = √(Vx² + Vy²)

V = √(5.00 m/s)² + (14.0 m/s)²

V ≈ √(25.0 + 196) ≈ √221 ≈ 14.9 m/s (to 3 significant figures)

D) The angle the ball makes with the horizon can be found using the tangent function:

θ = tan^(-1)(Vy / Vx)

θ = tan^(-1)(14.0 m/s / 5.00 m/s)

θ ≈ tan^(-1)(2.80)

θ ≈ 70.4 degrees (to 3 significant figures)

Therefore: A) The time the ball is in the air is approximately 1.43 seconds. B) The magnitude of the final velocity in the y-direction is approximately 14.0 m/s. C) The magnitude of the total final velocity is approximately 14.9 m/s. D) The angle the ball makes with the horizon is approximately 70.4 degrees