Slide 1 : INTRODUCTION &
RECTILINEAR KINEMATICS: CONTINUOUS MOTION Today’s Objectives:
Students will be able to:
Find the kinematic quantities (position, displacement, velocity, and acceleration) of a particle traveling along a straight path. In-Class Activities:
• Check Homework
• Reading Quiz
• Applications
• Relations between s(t), v(t), and a(t) for general
rectilinear motion.
• Relations between s(t), v(t),
and a(t) when acceleration is constant.
• Concept Quiz
• Group Problem Solving
• Attention Quiz
Slide 2 : READING QUIZ 1. In dynamics, a particle is assumed to have _________.
A) both translation and rotational motions
B) only a mass
C) a mass but the size and shape cannot be neglected
D) no mass or size or shape, it is just a point 2. The average speed is defined as __________.
A) Dr/Dt B) Ds/Dt
C) sT/Dt D) None of the above.
Slide 3 : APPLICATIONS The motion of large objects, such as rockets, airplanes, or cars, can often be analyzed as if they were particles.
Why? If we measure the altitude of this rocket as a function of time, how can we determine its velocity and acceleration?
Slide 4 : APPLICATIONS (continued) A sports car travels along a straight road.
Can we treat the car as a particle? If the car accelerates at a constant rate, how can we determine its position and velocity at some instant?
Slide 5 : An Overview of Mechanics
Slide 6 : RECTILINEAR KINEMATICS: CONTINIOUS MOTION
(Section 12.2) A particle travels along a straight-line path defined by the coordinate axis s. The total distance traveled by the particle, sT, is a positive scalar that represents the total length of the path over which the particle travels. The position of the particle at any instant, relative to the origin, O, is defined by the position vector r, or the scalar s. Scalar s can be positive or negative. Typical units for r and s are meters (m) or feet (ft).
Slide 7 : VELOCITY Velocity is a measure of the rate of change in the position of a particle. It is a vector quantity (it has both magnitude and direction). The magnitude of the velocity is called speed, with units of m/s or ft/s.
Slide 8 : ACCELERATION Acceleration is the rate of change in the velocity of a particle. It is a vector quantity. Typical units are m/s2 or ft/s2. As the book indicates, the derivative equations for velocity and acceleration can be manipulated to get a ds = v dv
Slide 9 : SUMMARY OF KINEMATIC RELATIONS:
RECTILINEAR MOTION • Differentiate position to get velocity and acceleration. v = ds/dt ; a = dv/dt or a = v dv/ds • Integrate acceleration for velocity and position. • Note that so and vo represent the initial position and velocity of the particle at t = 0.
Slide 10 : CONSTANT ACCELERATION The three kinematic equations can be integrated for the special case when acceleration is constant (a = ac) to obtain very useful equations. A common example of constant acceleration is gravity; i.e., a body freely falling toward earth. In this case, ac = g = 9.81 m/s2 = 32.2 ft/s2 downward. These equations are:
Slide 11 : EXAMPLE Plan: Establish the positive coordinate, s, in the direction the particle is traveling. Since the velocity is given as a function of time, take a derivative of it to calculate the acceleration. Conversely, integrate the velocity function to calculate the position. Given: A particle travels along a straight line to the right with a velocity of v = ( 4 t – 3 t2 ) m/s where t is in seconds. Also, s = 0 when t = 0. Find: The position and acceleration of the particle
when t = 4 s.
Slide 12 : EXAMPLE (continued) Solution: 1) Take a derivative of the velocity to determine the acceleration.
a = dv / dt = d(4 t – 3 t2) / dt =4 – 6 t
=> a = – 20 m/s2 (or in the ? direction) when t = 4 s
Slide 13 : CONCEPT QUIZ
Slide 14 : GROUP PROBLEM SOLVING Given: Ball A is released from rest at a height of 40 ft at the same time that ball B is thrown upward, 5 ft from the ground. The balls pass one another at a height of 20 ft. Find: The speed at which ball B was thrown upward. Plan: Both balls experience a constant downward acceleration of 32.2 ft/s2 due to gravity. Apply the formulas for constant acceleration, with ac = -32.2 ft/s2.
Slide 15 : GROUP PROBLEM SOLVING (continued) Solution: 1) First consider ball A. With the origin defined at the ground, ball A is released from rest ((vA)o = 0) at a height of 40 ft
((sA )o = 40 ft). Calculate the time required for ball A to drop to 20 ft (sA = 20 ft) using a position equation. sA = (sA )o + (vA)o t + (1/2) ac t2
So,
20 ft = 40 ft + (0)(t) + (1/2)(-32.2)(t2) => t = 1.115 s
Slide 16 : GROUP PROBLEM SOLVING (continued) Solution: 2) Now consider ball B. It is throw upward from a height of 5 ft ((sB)o = 5 ft). It must reach a height of 20 ft (sB = 20 ft) at the same time ball A reaches this height (t = 1.115 s). Apply the position equation again to ball B using t = 1.115s. sB = (sB)o + (vB)ot + (1/2) ac t2
So,
20 ft = 5 + (vB)o(1.115) + (1/2)(-32.2)(1.115)2
=> (vB)o = 31.4 ft/s
Slide 17 : ATTENTION QUIZ 2. A particle is moving with an initial velocity of v = 12 ft/s and constant acceleration of 3.78 ft/s2 in the same direction as the velocity. Determine the distance the particle has traveled when the velocity reaches 30 ft/s.
A) 50 ft B) 100 ft
C) 150 ft D) 200 ft
Slide 18 : End of the Lecture Let Learning Continue