Height = distance * tan(angle) = 40 * (1/√3) = 40/√3 = 20√3 meters.

Using tan(60°) = height/50, we have √3 = height/50. Therefore, height = 50√3 ≈ 86.6 m.

Distance = height / tan(angle) = 200 / (1/√3) = 200√3 m.

Distance = height / tan(angle) = 80 / tan(45°) = 80 meters.

Using tan(30°) = height/distance, we have 1/√3 = 80/distance. Therefore, distance = 80√3 ≈ 138.56 m.

Using tan(θ) = height/distance, we have tan(θ) = 15/20. Therefore, θ = tan⁻¹(0.75) which is approximately 36.87 degrees.

Using tan(60°) = height/distance, we have distance = height/tan(60°) = 40/√3 = 20√3 m.

Height = distance * tan(angle) = 10 * tan(45°) = 10 meters.

Using tan(60°) = height/10, we have √3 = height/10. Therefore, height = 10√3 m.

Using tan(60°) = height/50, we have √3 = height/50. Therefore, height = 50√3 ≈ 86.6 m.

Using tan(45°) = height/10, we have 1 = height/10. Therefore, height = 10 m.

The change in gravitational potential energy when a mass m is lifted to a height h is given by ΔU = mgh.

Damping ratio (ζ) = c / (2√(mk)) = 0.3 / (2√(1*4)) = 0.3 / 4 = 0.075.

The frequency of a mass-spring system is inversely proportional to the square root of the mass. Increasing the mass decreases the frequency.

Using the formula f = (1/2π)√(k/m), we can rearrange to find k = (2πf)²m. Thus, k = (2π(2))²(0.5) = 16 N/m.

Angular frequency (ω) = 2πf = 2π(3) = 6 rad/s.

Angular frequency ω = 2πf = 2π(3) = 6 rad/s.

The time period T is the reciprocal of frequency f. Thus, T = 1/f = 1/5 = 0.2 s.

Maximum speed v_max = ωA, where A is the amplitude. We have ω = √(k/m). Here, k = 100 N/m and m = 1 kg (assuming). Thus, A = v_max/ω = 4/10 = 0.4 m.

Angular frequency ω = 2π/T = 2π/2 = π rad/s ≈ 3.14 rad/s.

Frequency (f) is the reciprocal of the period (T). f = 1/T = 1/2 = 0.5 Hz.

Uncertainty in force = a * (uncertainty in mass) = 9.8 * 0.3 = 2.94 N.

Uncertainty in weight = g * (uncertainty in mass) = 9.8 * 0.1 = ±0.98 N, rounded to ±1 N.

The restoring force in simple harmonic motion is given by Hooke's law, which states that the force is proportional to the displacement and acts in the opposite direction. Therefore, the restoring force is -kx.

The time period T of a mass-spring system in simple harmonic motion is given by T = 2π√(m/k).

The angular frequency ω is given by ω = √(k/m).

Tension T = mv²/r. If r is halved, T doubles for constant speed.

At the highest point, the centripetal force is provided by the weight. Minimum speed = √(g*r).

At the highest point, T + mg = mv²/r, thus T = mg - mv²/r.

At the highest point, the centripetal force is provided by the weight of the mass, so T + mg = mv²/r. For T = 0, mg = mv²/r.