Concept explainers
A loaded railroad car is rolling at a constant velocity when it couples with a spring and dashpot bumper system. After the coupling, the motion of the car is defined by the relation x = 60e−4.8t sin 16t, where x and t are expressed in millimeters and seconds, respectively. Determine the position, the velocity, and the acceleration of the railroad car when (a) t = 0, (b) t = 0.3 s.
Fig. P11.4
(a)
The position (x), velocity (v), and acceleration (a) of the car when t is 0 seconds.
Answer to Problem 11.4P
The position (x), velocity (v) and acceleration (a) are
Explanation of Solution
Given information:
The function of time is
Calculation:
Write the relation for the motion of car:
Here, x is position of car and t is time.
Calculate the position (x) of the car when t is
Substitute
Calculate the velocity (v) of the car when t is 0 sec.
Differentiate Equation (1) with respect to time.
First derivative of position is equal to the velocity of the car. Rewrite the above equation as given below:
Here, v is velocity of the car.
Calculate the velocity (v):
Substitute 0 sec for t in Equation (2).
Calculate the acceleration (a) of the car when t is 0 sec.
Differentiate Equation (2) with respect to time.
First derivative of velocity is equal to the acceleration of the car.
Rewrite Equation (3),
Calculate the acceleration (a) of the car:
Substitute 0 sec for t in Equation (4).
Therefore, the position (x), velocity (v) and acceleration (a) are
(b)
The position (x), velocity (v), and acceleration (a) of the car when t is 0.3 seconds.
Answer to Problem 11.4P
Therefore, the position (x), velocity (v) and acceleration (a) are
Explanation of Solution
Given information:
The function of time is
Calculation:
Write the relation for the motion of car as given below:
Here, position of car is x in mm and time is t in seconds.
Calculate the position (x) of the car when t is
Substitute 0.3s for t in Equation (1).
Calculate the velocity (v) of the car when t is
Substitute 0.3s for t in Equation (3).
Calculate the acceleration (a) of the car when t is
Substitute 0.3s for t in Equation (3).
Therefore, the position (x), velocity (v) and acceleration (a) are
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Chapter 11 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
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