Tennessee State University Dr. Armwood Summer 2017 Dynamics

Question

Tennessee State University Dr Armwood Summer 20171 Dynamics Computer Tennessee State University Dr Armwood Summer 20171 Dynamics Computer Problems 1 & 2 Computer Problems Due: 08/10/. 1. A projectile enters a resisting medium at x=0 with an initial velocity v0 = 300 m/s and travels a distance d before coming to rest. The velocity of the projectile is defined by the relation v = v0 – 2700x, where v is expressed in meters/second and x in meters. a. Determine the distance d and derive expressions for the position, velocity, and acceleration of the projectile as functions of time. b. Plot the position, velocity, and acceleration of the projectile as functions of time from the time the projectile enters the medium until it penetrates a distance 0.99d. 2. The disk shown has a constant angular velocity of 500 rpm counterclockwise. Knowing that rod BD is 10 in. long, use computational software to determine and plot, for values of θ from 0–360°, (a) the velocity of collar D, (b) the angular velocity of rod BD, (c) determine the two values of θ for which the speed of collar D is zero.

Dr. Jack HW Helper · Accepted Answer

Introduction The analysis of projectile motion in resisting media and rotational systems are fundamental topics in dynamics. This paper addresses two problems: the first involves deriving the motion parameters of a projectile subjected to a linear resistive force, and the second involves kinematic analysis of a rotating disk with a connecting rod, focusing on velocities and angular velocities. Understanding the mathematical relations governing these systems enhances our ability to model real-world applications, such as ballistics and machinery design. Problem 1: Projectile Motion in a Resisting Medium The projectile enters the medium at an initial velocity v₀ = 300 m/s at position x = 0. The velocity decline is linear with respect to distance traveled, governed by v = v₀ – 2700x. The goal is to determine the total stopping distance d and the associated position, velocity, and acceleration functions relative to time. Determining the stopping distance d At the stopping point, the velocity becomes zero, so setting v = 0: 0 = v₀ – 2700d d = v₀ / 2700 = 300 / 2700 ≈ 0.1111 meters Deriving position, velocity, and acceleration functions of time Given v = v₀ – 2700x, and knowing that velocity v = dx/dt, we can write: dx/dt = v₀ – 2700x This is a first-order linear differential equation: dx/dt + 2700x = v₀ Solving using integrating factors: μ(t) = e^{2700t} d/dt [x e^{2700t}] = v₀ e^{2700t} Integrating: x e^{2700t} = (v₀ / 2700) e^{2700t} + C At t=0, x=0, so: 0 = (v₀ / 2700) + C → C = - (v₀ / 2700) Thus, the position as a function of time: x(t) = (v₀ / 2700) [1 – e^{-2700t}] Velocity as a function of time: v(t) = dx/dt = v₀ e^{-2700t} Acceleration: a(t) = dv/dt = -2700 v₀ e^{-2700t} Time to stop and plots Time to reach distance d: t_d = (1/2700) * ln(v₀ / (v₀ – 2700d)) Since v₀ – 2700d = 0, t_d approaches infinity; in real frequency, it asymptotically approaches stopping, but practically, the projectile stops at distance d. Plots of x(t), v(t), and a(t) from t=0 until the pr

Tennessee State University Dr. Armwood Summer 2017 Dynamics ✓ Solved

Tennessee State University Dr Armwood Summer 20171 Dynamics Computer

Sample Paper For Above instruction

Introduction

Problem 1: Projectile Motion in a Resisting Medium

Determining the stopping distance d

Deriving position, velocity, and acceleration functions of time

Time to stop and plots

Problem 2: Rotational Kinematics of a Disk with Connecting Rod

Conversion of units

Analysis of velocities and angular velocities

Conclusion

References