Calculating Time and Acceleration of a Toy Submarine
How much time has passed when the submarine's speed reaches 6.0×101% of its terminal value?
Given data: Initial mass (m), radius of circular track (r), constant propulsion force (F), proportionality factor for drag force (k)
What is the magnitude of the submarine's acceleration at this time?
Given data: Submarine's speed equation
Answer:
To find the time taken for the submarine's speed to reach a certain value and the magnitude of its acceleration at that time, we can use the provided data and equations.
To find the time taken for the submarine's speed to reach 6.0×101% of its terminal value, we can use the equation: t = (1/m)ln(1/(1−v/vt)), where t is the time, v is the speed, and vt is the terminal value of the speed.
Plugging in the given values, we get: t = (1/m)ln(1/(1−6.0×101/(100vt))).
To find the magnitude of the submarine's acceleration at this time, we can use the equation: a = (d/dt)v = (d/dt)(vt(1−e^(−t/m))).
Differentiating this equation with respect to time and plugging in the values, we can find the magnitude of the acceleration.