course Phy 121
At clock time t = 7 sec, a ball rolling straight down a hill is moving at 6 m/s and is 47 m from the top of the hill. It accelerates uniformly at a rate of .6 m/s/s until clock time t = 17 sec.
What is its velocity at this point: from t=7s to t=17sec, the velocity at t-17 is .6m/s/s for 10s = 6m.
.6 m/s/s is acceleration, which is the rate of change of velocity with respect to clock time. You cannot use this quantity to find a distance or displacement. Your answer, 6 meters, is a distance or a displacement.
What is the definition of rate of change of velocity with respect to clock time?
This definition includes three quantities, including the rate itself. Which two quantities are you given?
Which quantity can you therefore find and how?
What is the value of this quantity, and what is its meaning?
How does the value of this quantity help you solve the given problem?
and what is its average velocity during this time? 6m/s difference / 2 = average velocity of 3m/s.
The object starts at 6 m/s and speeds up. Its velocity is never less than 6 m/s. So its average velocity cannot be less than 6 m/s. Your answer of 3 m/s is not consistent with this reasoning.
Average velocity cannot be found using the difference in the two velocities.
What is the definition of average velocity?
What are the initial and final velocities of this object on this interval?
What therefore is the average velocity on this interval?
How far is it from the starting point at t = 17 sec? Average velocity of 3m/s for 10sec = 30m distance change from t=7s to t=17s. The distance from the top of the hill is 47m (the distance at t=7 plus the distance from t=7s + t=17s, therefore, 47m + 30m = 77m from the top of the hill.
This reasoning is good, but to get the correct answer you have to use a corrected average velocity.
You haven't quite gotten to the point where you consistently distinguish among initial velocity, final velocity, average velocity and change in velocity. You need to be very clear on the definitions of average velocity as average rate of change of position with respect to clock time, and average acceleration as average rate of change of velocity with respect to clock time.
These are not trivial concepts and you're making good progress. Try to be very careful to think in terms of the definitions, and I believe everything will fall into place for you.