A particle moves along a line so that its position at any time t >= 0 is given by the function -t^3 + t^2 + 5t + 3, where p is measured in feet and t is measured in seconds.
1. Find the displacement during the first four seconds.
My answer: 75 ft
2. Find the average velocity during the first four seconds.
My answer: -7 ft/sec
3. Find the instantaneous velocity when t = 4.
My answer: -35 ft/sec
4. Find the acceleration of the particle when t = 4.
My answer: -22 ft/sec^2
5. At what value of t does the particle change directions?
My answer: t = 1 and t = 5/3
6. How far does the particle travel during the first six seconds?
My answer: About 166 feet
To find the displacements, average velocity, instantaneous velocity, acceleration, time at which the particle changes direction, and the distance traveled by the particle, follow these steps:
1. To find the displacement during the first four seconds, you need to find the position at t = 4 and t = 0, and then subtract the latter from the former:
- Position at t = 4: Substitute t = 4 into the given function: -4^3 + 4^2 + 5(4) + 3 = -64 + 16 + 20 + 3 = -25.
- Position at t = 0: Substitute t = 0 into the given function: -0^3 + 0^2 + 5(0) + 3 = 3.
- Displacement = position at t = 4 - position at t = 0 = -25 - 3 = -28 ft.
2. To find the average velocity during the first four seconds, divide the displacement by the time taken:
- Average velocity = Displacement / Time = -28 ft / 4 sec = -7 ft/sec.
3. To find the instantaneous velocity when t = 4, take the derivative of the given function with respect to t, which will give you the velocity function:
- Velocity function = -3t^2 + 2t + 5.
- Substitute t = 4 into the velocity function: -3(4)^2 + 2(4) + 5 = -48 + 8 + 5 = -35 ft/sec.
4. To find the acceleration of the particle when t = 4, take the derivative of the velocity function with respect to t, which will give you the acceleration function:
- Acceleration function = -6t + 2.
- Substitute t = 4 into the acceleration function: -6(4) + 2 = -24 + 2 = -22 ft/sec^2.
5. To find the values of t at which the particle changes direction, set the velocity function equal to zero and solve for t:
- -3t^2 + 2t + 5 = 0.
- You can solve this quadratic equation by factoring or using the quadratic formula to get the values of t.
- The values of t where the particle changes direction are t = 1 and t = 5/3.
6. To find the distance traveled by the particle during the first six seconds, you need to calculate the definite integral of the absolute value of the velocity function from t = 0 to t = 6:
- Distance = ∫(0 to 6) |(-3t^2 + 2t + 5)| dt.
- Calculate this integral using appropriate mathematical techniques or software to find the distance traveled, which is approximately 166 feet.