Final answer:
To find the position and velocity of an object moving along a straight line, we use the equations of motion. For part a, the velocity function is v(t) = -e⁻ᵗ + 61 and the position function is s(t) = e⁻ᵗ + 61t + 39. For part b, the velocity function is v(t) = -10/(t+2) + 30 and the position function is s(t) = -10ln|t+2| + 30t + 16.
Step-by-step explanation:
In order to find the position and velocity of an object moving along a straight line, we need to use the equations of motion.
For part a, we can start by finding the velocity function. We have a(t) = e⁻ᵗ and v(0) = 60. We can integrate the acceleration function to get the velocity function: v(t) = ∫e⁻ᵗdt = -e⁻ᵗ + C. Using the initial velocity condition v(0) = 60, we can solve for the constant C: 60 = -e⁻⁰ + C, C = 61. Therefore, the velocity function is v(t) = -e⁻ᵗ + 61.
To find the position function, we integrate the velocity function: s(t) = ∫(-e⁻ᵗ + 61)dt = e⁻ᵗ + 61t + D. Using the initial position condition s(0) = 40, we can solve for the constant D: 40 = e⁻⁰ + 61(0) + D, D = 39. Therefore, the position function is s(t) = e⁻ᵗ + 61t + 39.
For part b, we follow the same steps. The acceleration function is a(t) = 20/(t+2)². Integrating the acceleration function gives us the velocity function: v(t) = ∫20/(t+2)²dt = -10/(t+2) + C. Using the initial velocity condition v(0) = 20, we can solve for the constant C: 20 = -10/(0+2) + C, C = 30. Therefore, the velocity function is v(t) = -10/(t+2) + 30.
Integrating the velocity function, we get the position function: s(t) = ∫(-10/(t+2) + 30)dt = -10ln|t+2| + 30t + D. Using the initial position condition s(0) = 10, we can solve for the constant D: 10 = -10ln|0+2| + 30(0) + D, D = 16. Therefore, the position function is s(t) = -10ln|t+2| + 30t + 16.