Question

A particle moves along a straight line. For ≥ 0 the velocity of the particle is given by () = ^(sin) − 1, and the position of the particle is given by (), a continuous differential function. It is known (0) = 3. a. For 2 ≤ ≤ 4, find all the values of for which the speed of the particle is equal to 0.5. b. Find the average acceleration of the particle for 1 ≤ ≤ 3.

Answer 1

To answer the particle motion problem, we set the absolute value of the velocity function equal to 0.5 and solved it within the specified range. For average acceleration, we found the derivative of velocity, evaluated it at the interval's endpoints, and divided the difference by 2.

Step-by-step explanation:

To solve the given particle motion problem, we first need to understand that speed is the absolute value of velocity, so the equation for 'speed = 0.5' is |v(t)| = 0.5. To find the average acceleration during the interval [1,3], we calculate the change in velocity divided by the change in time. This is the same as the value of the derivative of the velocity function at 't = 3' minus the value of the derivative at 't = 1' divided by 2.

To be more specific:

• To find the values of 't' for which 'speed = 0.5', we set: |t^sin(t) - 1| = 0.5. We solve this equation for the interval [2,4].
• To find the average acceleration over the interval [1,3], we differentiate the velocity function to get the acceleration function a(t), evaluate a(3) and a(1), subtract these two values and divide the result by 2.

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