An object moves in a straight line. Its instantaneous velocity is negative and its instantaneous. Which of the following can be determined about the object's motion- A) The object is speeding up B) The object is slowing down C) The object will come to rest D) The object will change direction (PLEASE ANSWER WITH EXPLANATION)
Answers
Explanation:
The quantity that tells us how fast an object is moving anywhere along its path is the instantaneous velocity, usually called simply velocity. It is the average velocity between two points on the path in the limit that the time (and therefore the displacement) between the two points approaches zero. To illustrate this idea mathematically, we need to express position x as a continuous function of t denoted by x(t). The expression for the average velocity between two points using this notation is
–
v
=
x
(
t
2
)
−
x
(
t
1
)
t
2
−
t
1
v–=x(t2)−x(t1)t2−t1. To find the instantaneous velocity at any position, we let
t
1
=
t
t1=t and
t
2
=
t
+
Δ
t
t2=t+Δt. After inserting these expressions into the equation for the average velocity and taking the limit as
Δ
t
→
0
Δt→0, we find the expression for the instantaneous velocity:
v
(
t
)
=
lim
Δ
t
→
0
x
(
t
+
Δ
t
)
−
x
(
t
)
Δ
t
=
d
x
(
t
)
d
t
.
v(t)=limΔt→0x(t+Δt)−x(t)Δt=dx(t)dt.
Instantaneous Velocity
The instantaneous velocity of an object is the limit of the average velocity as the elapsed time approaches zero, or the derivative of x with respect to t:
v
(
t
)
=
d
d
t
x
(
t
)
.
v(t)=ddtx(t).
Like average velocity, instantaneous velocity is a vector with dimension of length per time. The instantaneous velocity at a specific time point
t
0
t0 is the rate of change of the position function, which is the slope of the position function
x
(
t
)
x(t) at
t
0
t0. (Figure) shows how the average velocity
–
v
=
Δ
x
Δ
t
v–=ΔxΔt between two times approaches the instantaneous velocity at
t
0
.
t0. The instantaneous velocity is shown at time
t
0
t0, which happens to be at the maximum of the position function. The slope of the position graph is zero at this point, and thus the instantaneous velocity is zero. At other times,
t
1
,
t
2
t1,t2, and so on, the instantaneous velocity is not zero because the slope of the position graph would be positive or negative. If the position function had a minimum, the slope of the position graph would also be zero, giving an instantaneous velocity of zero there as well. Thus, the zeros of the velocity function give the minimum and maximum of the position function.
Graph shows position plotted versus time. Position increases from t1 to t2 and reaches maximum at t0. It decreases to at and continues to decrease at t4. The slope of the tangent line at t0 is indicated as the instantaneous velocity.