a particle moves along a straight line such that its position is defined by s=(t^2-3t^2=2)m.determine the average velocity, the average speed, and the acceleration of the particle when t=4s.
Unusual wording of this question
When they ask for "average" velocity and acceleration, usually there is a time interval involved.
In your case , we are given t = 4, a specific time.
I would expect to be asked for the "instantaneous" velocity and acceleration
method:
if s = t^2 - 3t^2 + 2 (I assume a typo, since the = and + on on the same key , also is one of those a cubic? )
v= derivative of s
a = derivatiave of v
sub in t = 4
To find the average velocity, average speed, and acceleration of the particle at t=4s, we need to differentiate the given position equation with respect to time.
Given: s = t^2 - 3t^2 + 2
1. Average Velocity:
The average velocity is the change in position with respect to time. To find it, we need to differentiate the position equation with respect to t.
ds/dt = 2t - 6t
Now we substitute t=4s into the derived equation:
V = ds/dt = 2(4) - 6(4) = 8 - 24 = -16 m/s
Therefore, the average velocity at t=4s is -16 m/s.
2. Average Speed:
The average speed is the total distance traveled divided by the total time taken. To find it, we need to calculate the total distance traveled at t=4s.
Let's find the displacement from t=0s to t=4s:
s(4) - s(0) = (4^2 - 3(4) + 2) - (0^2 - 3(0) + 2)
= (16 - 12 + 2) - (0 - 0 + 2)
= 6 - 2
= 4m
The total distance traveled is 4m. Since the total time taken is 4s, the average speed is:
Average Speed = Total Distance / Total Time = 4m / 4s = 1m/s
Therefore, the average speed at t=4s is 1 m/s.
3. Acceleration:
Acceleration is the rate of change of velocity. To find it, we differentiate the velocity equation with respect to time.
dV/dt = d(2t - 6t) / dt = 2 - 6 = -4 m/s^2
Therefore, the acceleration at t=4s is -4 m/s^2.
In summary:
- The average velocity at t=4s is -16 m/s (negative indicates motion in the opposite direction).
- The average speed at t=4s is 1 m/s (scalar value with no direction).
- The acceleration at t=4s is -4 m/s^2 (negative indicates deceleration).