15-94 Chapter 15: Homework Problems

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1 15-94 Chapter 15: Homework Problems 13.1 Rectilinear Motion (a) The displacement of a particle is given by s = At 3 Bt 2 Ct 50m. If A = 1m/s 3, B = 2m/s 2, and C = 3m/s, plot the displacement, velocity, and acceleration for the first 6 seconds. (b) The velocity of a particle is given by v x = x and s = 1m at t = 0, determine the position as a function of time. (c) The acceleration of a particle is given by a x = 2v x, and x = 0 and v x = 1m/s at t = 0. Determine the velocity as a function of time Motorcycles (a) A motorcyclist moves at a speed of 35 m/s down a straight road and sees traffic ahead. If the rider applies the brakes lightly and decelerates at a constant rate of 5m/s 2, how far does the motorcycle travel before it stops. (b) In the same scenario, how far does the motorcycle travel if it decelerates at a rate of (5m/s 3 )t? 13.3 The first appearance of Superman was in Action Comics #1 in Created by writer Jerry Siegel and artist Joe Shuster, his powers granted him the ability to leap tall buildings in a single bound, made him faster than a speeding bullet, and he was more powerful than a locomotive. Lets say that this early version of Superman cannot fly and wants to leap to the top of a tall building (400 ft.) in a single bound. If we model him as a particle and he takes off at an angle of 80 o, what is the minimum initial speed required to get him to the top of the building? How far away from the base of the building should he take off? You may neglect drag on this problem and assume that the only acceleration is that due to gravity.

2 Chapter 15: Homework Problems Find the world record for the male and female high jump. If the high jumper is modeled as a particle (you may assume that the jumper is a particle with their mass concentrated at their midsection = 1 of their height), determine the take off velocity required to just 2 clear the bar in each case. Assume that the takeoff point is 2.5 ft. in front of the bar. You may neglect drag for this problem and assume that the only acceleration is that due to gravity.

3 15-96 Chapter 15: Homework Problems 13.5 A ball moves along a curve from A to B whose path is given by y = (12 x 2 ) where x and y are measured in feet. At x = 1.0ft., the ball is moving in the x-direction at a constant rate of 2 ft./s. Draw the basis vectors for path coordinates on the figure below. What is the velocity vector when x = 1.0f t.? What is the acceleration vector when x = 1.0ft.? Calculate the radius of curvature and rate of change of speed at the same point. y 12 x

4 Chapter 15: Homework Problems A bead is constrained to move along a path given by y = 0.5x 3 where x and y are measured in meters. At x = 5m, the bead is moving in the x direction at a constant rate of 0.25 m/s. Draw the basis vectors for path coordinates on the figure below. What are the velocity and acceleration vectors when x = 5m? Calculate the radius of curvature and rate of change of speed at the same point.

5 15-98 Chapter 15: Homework Problems 13.7 A car traveling around a curve at a speed of 100 km/h is decelerating at a constant rate of 20m/s 2. If the radius of curvature is 550 m, calculate the magnitude of the total acceleration at this instant. vn vt ρ

6 Chapter 15: Homework Problems The clock is ticking and there is one last chance. Someone has to throw it the length of the field. If b = 20yards and the ball leaves the quarterback s hand at an angle of θ = 34 o, what does the initial velocity have to be in order for the ball to reach point B? Calculate the radius of curvature and rate of change of speed at the apex of the ball s flight A c b B a B c θ B A 13 For those that aren t familiar with American football, kudos, you are saving yourself heartbreak. The numbers on the diagram indicate distance in yards.

7 Chapter 15: Homework Problems 13.9 Suppose that a particle has a radial position, r = At 3 Bt 2 Ct 50m and constant angular velocity (i.e. constant θ) 0.45 rad/s. Note that θ = 0 at t = 0. If A = 1m/s 3, B = 2m/s 2, and C = 3m/s, determine the position, velocity, and acceleration at t = 3.5s. Sketch the path from t = 0 to 3.5 s and sketch the basis vectors ( e r and e θ ) at t = 3.5s.

8 Chapter 15: Homework Problems The radius of the Earth is approximately 3,950 miles. Consider a satellite in a Clarke orbit 14. If the satellite has an altitude above the Earth of 22,236 miles and appears stationary from the Earth below, calculate the velocity and acceleration of the satellite. Note that the acceleration due to gravity is g = (9.81m/s 2 ) R 2 (R+r) 2. R r A 14 A Clarke orbit is named for famed Science Fiction writer and futurist, Arthur C. Clarke. It is another name for a geosynchronous orbit.

9 Chapter 15: Homework Problems The housing of the hydraulic cylinder has a length a = 0.75m and the end of the actuator is moving inward at a constant speed of v r = 0.25m/s. If θ = 30 o and is decreasing at a constant rate of 5 o /s, determine the velocity and acceleration of point B. The value of b = 0.5m. vr b B a A θ,ω

10 Chapter 15: Homework Problems All three cars are moving with the same speed (45 m.p.h.). Car C is traveling around a curve with constant radius r = 100ft. and, at this instant, is located at θ = 35 o. Determine the following quantities: v A v B v A v C a A a B a A a C va y x A C B vb θ r vc

11 Chapter 15: Homework Problems A rocket is fired straight up and is tracked by a RADAR 15 system located at the origin. At one particular instant, the RADAR system notes that the radius, r, is 100,000 ft., θ = 70 o, θ = 0.01rad/s, and r = 85ft./s 2. Calculate the speed and the magnitude of the acceleration. USA y r θ x 15 RADAR stands for RAdio Detection And Ranging.

12 Chapter 15: Homework Problems This time our rocket is fired straight up, but follows a curved path. At one particular instant, the RADAR system notes that the radius, r, is 100,000 ft., ṙ = 2300ft./s, θ = 0.01rad/s, and θ = rad/s 2. Sketch the flight path for the rocket. Calculate the radius of curvature at this instant. You may assume that θ = 75 o and r = 20ft./s 2. USA y r θ x

13 Chapter 15: Homework Problems The Ferris Wheel rotates with a constant angular velocity of 5 o /s and a car is driving away at a constant speed of approximately 25 m/s. Suppose that, the rider starts at θ = 0 o and the car is directly below them (also y = 5m, r = 20m). At what time would the car and the Ferris Wheel passenger be exactly 200 m apart in straight line distance? What is their relative acceleration at this time? ω A θ r y x B v

14 Chapter 15: Homework Problems Pin P travels on a wire whose shape is given by y = 1 2 x2. Pin P is also constrained to move within a slot in guide B (it can only move in the x-direction) where the guide has a constant speed of v B = 10ft./s to the right. At the position shown, x = 2ft. For this position, find the velocity and acceleration of P. Write you r answers as vectors in Cartesian coordinates. Sketch the velocity and acceleration vectors in the figure. Sketch the unit normal vectors, e t and e n. Determine the rate of change of speed of P.

15 Chapter 15: Homework Problems A fireplane is at the bottom of a circular loop. The loop s radius is 950 ft. and its center is directly above point O. The plane is traveling at a constant speed. At the instant shown, it is 100 ft. above point O and traveling at 200 ft./s. Sketch the unit vectors for path coordinates and polar coordinates. Sketch the velocity and acceleration vectors. Calculate the total acceleration of the plane in vector form (using path coordinates). Calculate ṙ and θ, r, and θ at the instant shown. If 100 gallons of water is dropped at that time, how far away from point O does it land? You may neglect the effects of wind resistance on the falling water. v A r h B x O θ

16 Chapter 15: Homework Problems A particle, P, travels within the x y plane along a path given by x = y3, where x and 3 y are given in feet. Draw the path of the particle. The x-component of the position for P is increasing at a constant rate of 9 ft./s. For the position of P corresponding to y = 3f t., determine the velocity and acceleration vectors of P in Cartesian coordinates, the rate of change of speed of P, and the radius of curvature.

15-110 Chapter 15: Homework Problems 13.19 Block A has an acceleration of 4m/s 2 to the right.

17 Chapter 15: Homework Problems Block A has an acceleration of 4m/s 2 to the right. What is the magnitude of the acceleration of block B? Is the acceleration directed toward the right or the left?

18 Chapter 15: Homework Problems The two blocks are connected by an inextensible cable. If block 2 moves downward at 3 m/s, determine the speed of block

19 Chapter 15: Homework Problems If W 2 is moving down and to the right at 3 m/s and accelerating at 0.5m/s 2, what are the velocity and acceleration of W 1? θ W2 W1

Addis Ababa University Addis Ababa Institute of Technology School Of Mechanical and Industrial Engineering Extension Division` Assignment 1

Addis Ababa University Addis Ababa Institute of Technology School Of Mechanical and Industrial Engineering Extension Division` Assignment 1 Assignment 1 1. Vehicle B is stopped at a traffic light, as shown in the figure. At the instant that the light turns green, vehicle B starts to accelerate at 0.9144m/s 2. At this time vehicle A is 91.44m