1. Chapter 2: Motion in One Dimension (examples and applications)
Physics 201, Lect. 4 Jan. 28, 2010
Chapter 2: Motion in One Dimension
(examples and applications)

2. Physics 103, Fall 2009, U. Wisconsin
“stopping distance”:
You were driving on a country road at an instantaneous velocity of 55 mph, east. You suddenly saw a family of ducks walking across the road 30 m ahead of you (the stopping distance).
You applied the break hard and stopped right in front of the ducks.
What was the acceleration you applied?
xi=0, xf=30 m,
vi = 55 mph = (55x1.609) 88.5 km/hr = 24.6 m/s, vf=0,
0 = vi2 + 2 a xf
- a = - vi2/2xf = - 10 m/s2
A large deceleration!
How to avoid this sharp change?
11/22/2018
Physics 103, Fall 2009, U. Wisconsin

3. Example 2-13, The Flying Cap (p 44)
A cap is tossed in the air. It starts at height y0 with a velocity v0y=14.7 m/s upward.
How long does it take for the cap to reach its highest point?
How high does the cap go?
©2008 by W.H. Freeman and Company

4. Example 2-13, The Flying Cap (p 44)
A cap is tossed in the air. It starts at height y0 with a velocity v0y=14.7 m/s upward.
How long does it take for the cap to reach its highest point?
Highest point is when the velocity is zero.
vy(t) = v0y + aytf  tf = (vy(t)-v0y)/ay
= ( m/s)/(-9.81 m/s2) = 1.50 s
How high does the cap go?
The cap goes distance: yf = v0ytf + (1/2)aytf2
= (14.7 m/s)(1.50 s) + (1/2)(-9.81 m/s2)(1.50 s)2
= – = 11.0 m.
©2008 by W.H. Freeman and Company

5. Example 2-13; Height and velocity of the flying cap
Height function of t
quadratic.
Velocity function
linear.
Acceleration = const.
©2008 by W.H. Freeman and Company

6. Getting velocity from acceleration by integration
Constant acceleration:
v(t) = v0 + a (t – t0)
(The only tricky part is putting in the limits of integration.)

7. Getting position from velocity by integration
Constant acceleration:
x(t) = x0 + v0 (t – t0) + (1/2) a (t – t0)2
(The only tricky part is putting in the limits of integration.)

8. The displacement is the integral of the velocity (the limit of the sum of areas of the rectangles and it can be negative!).
©2008 by W.H. Freeman and Company

9. For constant acceleration, as we saw before:
Area = x(t)
= v0x t + (1/2) a t2
©2008 by W.H. Freeman and Company

10. Problem 2.31
Reginald and Josie go the same distance within the same time. What is relationship between vJ and vmax? vJ
©2008 by W.H. Freeman and Company

11. Problem 2.31
Reginald and Josie go the same distance within the same time. What is relationship between vJ and vmax? vJ
Distance traveled is area under v-versus-t curve.
For Josie, this area is vJtf. For Reginald, this area is tf vmax / 2. The areas are equal when vmax=2vJ.
©2008 by W.H. Freeman and Company

12. Problem 2.106 write algebraic expressions for x(t), vx(t), and ax(t)
©2008 by W.H. Freeman and Company

13. Problem 2.106 write algebraic expressions for x(t), vx(t), and ax(t)
Reading from graph, vx(t) = v0 + At, with v0=50 m/s and A=-10 m/s2.
Acceleration is derivative of velocity: ax(t)=A=-10m/s2.
Position is integral of velocity: x(t) = x(0)+v0t + ½At2 = 50t - 5t2.
©2008 by W.H. Freeman and Company

14. Motion with time-varying acceleration
An object undergoes a linearly increasing acceleration a(t) = 10 t m/s2 along a straight line. It starts at the origin x(0)=0 with a velocity of 2 m/s. Where is its position at time t=4 seconds?

15. Motion with time-varying acceleration
An object undergoes a linearly increasing acceleration a(t) = 10 t m/s2 along a straight line. It starts at the origin x(0)=0 with a velocity of 2 m/s. What is its position at time t=4 seconds?
In this problem the acceleration is NOT constant. So must do the integrals explicitly.

16. Which graph of v versus t best describes the motion of a particle with positive velocity and negative acceleration?

17. Which graph of v versus t best describes the motion of a particle with positive velocity and negative acceleration?

18. A car accelerates uniformly from a velocity of 10 km/h to 30 km/h in one minute. Which graph best describes the motion of the car?

19. A car accelerates uniformly from a velocity of 10 km/h to 30 km/h in one minute. Which graph best describes the motion of the car?

20. The acceleration of a vehicle is given by a(t) = At where A is a constant. Its velocity as a function of time is (vo is a constant)

21. The acceleration of a vehicle is given by a(t) = At where A is a constant. Its velocity as a function of time is (vo is a constant)

22. Conceptual question: velocity and acceleration
A ball is thrown vertically upward. At the very top of its trajectory, which of the following statements is true:
a. velocity is zero and acceleration is zero b. velocity is not zero and acceleration is zero c. velocity is zero and acceleration is not zero d. velocity is not zero and acceleration is not zero

23. Conceptual question: velocity and acceleration
A ball is thrown vertically upward. At the very top of its trajectory, which of the following statements is true:
a. velocity is zero and acceleration is zero b. velocity is not zero and acceleration is zero c. velocity is zero and acceleration is not zero d. velocity is not zero and acceleration is not zero
correct
At the top of the path, the velocity of the ball is zero, so that it doesn’t go any higher.
The velocity at the top is still changing, and the acceleration is the change in velocity. Just because the velocity is zero does not mean that it is not changing. So the acceleration is NOT zero.
Acceleration is due to gravity, and is constant, and
always a downward-pointing vector.