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rue7
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Hello! I have a problem which is solvable using simpler methods, but I'm trying to use it as a bridge to understanding how to do these problems in a more rigorous setting.
A train slows down as it rounds a sharp horizontal turn, slowing from 90 km/hr to 50 km/hr in the 15 s that it takes to round the bend. The radius of the curve is 150 m. Compute the acceleration at the moment the train speed reaches 50 km/hr. Assume it continues to slow down at this time at the same rate.
First Attempt
My first attempt involved starting with a position vector
r(t) = acos(t)i + asin(t)j
where a = the radius of the curve.
However, when I took the derivative of this my attempt quickly crumbled, because:
r'(t) = v(t) = -asin(t)i + acos(t)j
in which:
|v(t)| = \sqrt{a^{2}sin^{2}(t) + a^{2}cos^{2}(t)} = a
The magnitude of the velocity vector = a at all times t? I knew this couldn't be true so I made my second attempt.
Second Attempt
This time I started with a tangent acceleration + radial acceleration vector:
a(t) = a_{t} T + a_{n}\kappa N
where T is the tangent unit vector, N is the normal (or radial) unit vector and \kappa is the curvature of the circle.
My failure here is either a lack of understanding in how to present the circular motion's acceleration or in my own ability to manipulate that vector back into a velocity vector and as well as a position vector.
Additional note:
As I said in the beginning, I know this problem is solvable by using formulas already derived using vectors and Calculus. In fact, this problem was taken from a book for a course that only requires Calculus 1. However, when I discovered how to solve Two-Dimensional (not rotational) Motion problems using derivatives and integrals rather than the "suvat" Equations of Motion beginning Physics students are taught to memorize, I immediately became curious how this can be done with Circular Motion Problems.
Many thanks if you read this far. Infinite thanks in advance for anyone who chooses to help me understand.
Homework Statement
A train slows down as it rounds a sharp horizontal turn, slowing from 90 km/hr to 50 km/hr in the 15 s that it takes to round the bend. The radius of the curve is 150 m. Compute the acceleration at the moment the train speed reaches 50 km/hr. Assume it continues to slow down at this time at the same rate.
Homework Equations
The Attempt at a Solution
First Attempt
My first attempt involved starting with a position vector
r(t) = acos(t)i + asin(t)j
where a = the radius of the curve.
However, when I took the derivative of this my attempt quickly crumbled, because:
r'(t) = v(t) = -asin(t)i + acos(t)j
in which:
|v(t)| = \sqrt{a^{2}sin^{2}(t) + a^{2}cos^{2}(t)} = a
The magnitude of the velocity vector = a at all times t? I knew this couldn't be true so I made my second attempt.
Second Attempt
This time I started with a tangent acceleration + radial acceleration vector:
a(t) = a_{t} T + a_{n}\kappa N
where T is the tangent unit vector, N is the normal (or radial) unit vector and \kappa is the curvature of the circle.
My failure here is either a lack of understanding in how to present the circular motion's acceleration or in my own ability to manipulate that vector back into a velocity vector and as well as a position vector.
Additional note:
As I said in the beginning, I know this problem is solvable by using formulas already derived using vectors and Calculus. In fact, this problem was taken from a book for a course that only requires Calculus 1. However, when I discovered how to solve Two-Dimensional (not rotational) Motion problems using derivatives and integrals rather than the "suvat" Equations of Motion beginning Physics students are taught to memorize, I immediately became curious how this can be done with Circular Motion Problems.
Many thanks if you read this far. Infinite thanks in advance for anyone who chooses to help me understand.