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A motorcycle daredevil plans to ride up a 2.0-m-high, 20∘ ramp, sail across a 10-m-wide pool filled with hungry crocodiles, and land at ground level on the other side. He has done this stunt many times and approaches it with confidence. Unfortunately, the motorcycle engine dies just as he starts up the ramp. He is going 10.4 m/s at that instant, and the rolling friction of his rubber tires (coefficient 0.02) is not negligible. Part A: Justify your answer by calculating the distance he travels through the air after leaving the end of the ramp.

Sagot :

Answer:

The motorcycle daredevil does not sail across the 10-m wide pool filled with hungry crocodiles

Explanation:

The given parameters are;

The height of the ramp = 2.0 m-high

The angle of elevation of the ramp, θ = 20°

The width of the pool = 10 m

The speed with which he is going when the engine fails = 10.4 m/s

The rolling friction between the rubber tires, μ = 0.02

μ·g·cos(θ) = (1/2)·(a)

a = (μ·g·cos(θ))/(1/2)

Where;

a = The acceleration down the plane

∴ a = 0.02 × 9.81 × cos(20°)/(1/2) ≈ 0.369

The length of the ramp, l = 2.0 m/(sin(20°)) ≈ 5.8476088 m

The velocity at the end of the ramp, v² = u² - 2·a·l

∴ v² = 10.4² - 2×(0.369 + 9.81 × sin(20°))×5.8476088 ≈ 64.60446471

v ≈ √(64.60446471) ≈ 8.04

∴ The velocity at the top of the ramp, v ≈ 8.04 m/s

The time it takes the motorcycle to reach ground level from the top of the ramp, t, is given as follows;

h = h₀ + u·t + (1/2)·a·t²

h₀ = 2, u = 8.04×sin(20°), a = -9.81

When h = 0 (ground level), we have;

∴ h = 2+ 8.04×sin(20°)·t - (1/2)×9.81×t²

0 ≈ 2.75·t - 4.905·t² + 2

2.75·t - 4.905·t² + 2 = 0

By using a graphing calculator, we have, t = 0.978 or t = -0.42

At ground level, the time of flight, t ≈ 0.978 s

The horizontal distance the motorcycle travels, before landing, x = u × cos(θ) × t

∴ x = 8.04×cos(20 Degrees) × 0.978 = 7.3889

The horizontal distance the motorcycle travels, before landing, x ≈ 7.3889 m

Therefore, given that the horizontal distance the motorcycle travels in the air 7.3889 m < 10 m, therefore the motorcycle daredevil does not sail across the 10-m wide pool filled with hungry crocodiles

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