Average Acceleration and Instantaneous Acceleration
When a particle's velocity changes from to in a time interval , its average acceleration during is
If we shrink to zero about some instant, then in the limit approaches the instantaneous acceleration (or acceleration) at that instant; that is,
If the velocity changes in either magnitude or direction (or both), the particle must have an acceleration.
We can write Eq. 4-16 in unit-vector form by substituting for v*from Eq. 4-11 to obtain
We can rewrite this as
where the scalar components of are
We can rewrite this as
where the scalar components of are
Sample Problem 4-4
For the rabbit in Sample Problems 4-2 and 4-3, find the acceleration at time s, in unit-vector notation and as a magnitude and an angle.
Solution: There are two Key Ideas here: (1) We can find the rabbit's acceleration by first finding the acceleration components. (2) We can find those components by taking derivatives of the rabbit's velocity components. Applying the first of Eqs. 4-18 to Eq. 4-13, we find the component of to be
Similarly, applying the second of Eqs. 4-18 to Eq. 4-14 yields the component as
We see that the acceleration does not vary with time (it is a constant) because the time variable does not appear in the expression for either acceleration component. Equation 4-17 then yields
which is shown superimposed on the rabbit's path in Fig. 4-8.
To get the magnitude and angle of , either we use a vector-capable calculator or we follow Eq. 3-6. For the magnitude we have
For the angle we have
For each description, determine whether the x and ó components of the puck's acceleration are constant, and whether the acceleration a* is constant.
However, this last result, which is displayed on a calculator, indicates that is directed to the right and downward in Fig. 4-8. Yet, we know from the components above that must be directed to the left and upward. To find the other angle that has the same tangent as - 35°, but which is not displayed on a calculator, we add 180°:
-35° + 180° = 145c
This is consistent with the components of . Note that has the same magnitude and direction throughout the rabbit's run because, as we noted previously, the acceleration is constant.
1. The position of an object moving only an -axis is given by , where is in meters and in seconds. (a) What is the position of the object at s? (b) What is the displacement’s between and s? (c) What is the average velocity for the time interval from and ?
2. (a) If particle’s position is given by , (where is in meters and in seconds), what its velocity at ? (b) Is it moving in positive or negative direction of just then? (c) What is its speed just then? (d) Is the speed larger or smaller at later time? (e) Is there ever an instant when the velocity is zero? (f) Is there a time after when the particle is moving in the negative direction of ?
3. The position of a particle moving along the axis is given in centimeters by , where is in seconds. Calculate (a) the average velocity during the time interval s to s; (b) the instantaneous velocity at s; (c) the instantaneous velocity at s; (d) the instantaneous velocity at s; and (e) the instantaneous velocity when the particle is midway between its positions at f s and s. (f) Graph versus and indicate your answers graphically.
4. A proton moves along the axis according to the equation , where is in meters and is in seconds. Calculate (a) average velocity of the proton during the first 3.0 s of its motion. (b) the instantaneous velocity of the proton at s, (c) the instantaneous acceleration of the proton at s.
5. An electron moving along the axis has a position given by m. where is in seconds. How far is the electron from the origin when it momentarily stops?
6. The position of a particle moving along the axis depends on the time according to the equation , where is in meters and in seconds, (a) What units must ñ and b have? Let there numerical values be 3.0 and 2.0, respectively. (b) At what time does the particle reach its maximum positive position? From and , (c) what distance does the particle move and (d) what is its displacement? At s what are (e) its velocity and (f) its acceleration?
The and coordinates of a particle at any time are given by and , where and are in meters and in seconds. The acceleration of the particle at s is
(A) zero (B) 8 m/ s2 (C) 20 m/s2 (D) 40 m·s2
2-34 A jet-propelled motorcycle starts from rest, moves in a straight line with constant acceleration, and covers a distance of 64 m in 4 s.
a)What is the final velocity?
b)How much time was required to cover half the total distance?
c) What is the distance covered in one-half the total time?
d)What is the velocity when half the total distance has been covered?
e) What is the velocity after one-half the total time?
f) When will the instantaneous velocity equal the average velocitv for the 0-to-4 s time interval?
Q. 2.09. If the displacement of a body is proportional to square of time, state whether the body is moving with uniform velocity or uniform acceleration.
Q. 2.10. If the distance traveled by a body in time is given by , then what will be the acceleration of the body?
Q. 2.11. An object is covering distance in direct proportion to , where is the time elapsed.
(a) What conclusions might you draw about acceleration? Is it constant? Increasing? Decreasing?
(b) What might you conclude about the force acting on the object?
2. The position of a particle is given by , where t is in seconds and the coefficients have the proper units for to be in meters.
(a) Find and of the particle.
(b) Find the magnitude and direction of at s.
[Ans. (a) ; ;(b) 12.37 m/s; 76°]
Section 2—5 Velocity and Coordinate by Integration
2-20 The motion of a particle along a straight line is described by the function
x = (6 m) + (5 m-s-2)/2 - (1 m-s-4)/4.
Assume that t is positive.
a) Find the position, velocity, and acceleration at time t = 2 s.
b)During what time interval is the velocity positive?
c) During what time interval is x positive?
d)What is the maximum positive velocity attained by the particle?
2—21 The acceleration of a motorcycle is given by a - (1.2 m-s-3)/ - (0.12 m-s-4)/2. It is at rest at the origin
at time t — 0.
Find its position and velocity as functions of time.
a) Calculate the maximum velocity it attains.
2-22 The acceleration of a bus is given by a = (2 m-s-3)t.
) If the bus's velocity at time t = 1 s is 5 m-s-1, what is its velocity at time ( = 2s?
b) If the bus's position at time t = 1 s is 6 m, what is its position at time t = 2 s?
56 The acceleration of an object suspended from a spring and oscillating vertically is , where is a constant and is the coordinate measured from the equilibrium position. Suppose that an object moving in this way is given an initial velocity at the coordinate .Find the expression for the velocity of the object as a function of its coordinate .
2-57 The motion of an object falling from rest in a resisting medium is described by the equation
where A and Â are constants. In terms of A and B, find
a) the initial acceleration;
b) the velocity at which the acceleration becomes zero (the terminal velocity).
c) Show that the velocity at any given t is given by
2-58 After the engine of a moving motorboat is cut off, the boat has an acceleration in the opposite direction to its velocity and directly proportional to the square of its velocity. That is, , where is constant.
a) Show that the magnitude of the velocity at a time after the engine is cut off is given by
b) Show that the distance x traveled in a time is
c) Show that the velocity after traveling a distance is
As a numerical example, suppose the engine is cut off when the velocity is 6 m/s, and that the velocity decreases to 3 m/s in a time of 15 s.
d) Find the numerical value of the constant , and the unit in which it is expressed.
e) Find the acceleration at the instant the engine is cut off.
f) Calculate x, v, and a at 10-s intervals for the first 60 s after the engine is cut off. Sketch graphs of x, v. and a as