A graph of velocity vs. time of a ship coming into a harbor is shown below. (a) Describe the motion of the ship based on the graph. (b)What would a graph of the ship’s acceleration look like?
(a) The ship moves at constant velocity and then begins to decelerate at a constant rate. At some point, its deceleration rate decreases. It maintains this lower deceleration rate until it stops moving.
(b) A graph of acceleration vs. time would show zero acceleration in the first leg, large and constant negative acceleration in the second leg, and constant negative acceleration.
Graphical solutions yield identical solutions to mathematical methods for deriving motion equations.
The slope of a graph of displacement
vs. time
is velocity
.
The slope of a graph of velocity
vs. time
graph is acceleration
.
Average velocity, instantaneous velocity, and acceleration can all be obtained by analyzing graphs.
Conceptual questions
(a) Explain how you can use the graph of position versus time in
[link] to describe the change in velocity over time. Identify (b) the time (
,
,
,
, or
) at which the instantaneous velocity is greatest, (c) the time at which it is zero, and (d) the time at which it is negative.
(a) Sketch a graph of velocity versus time corresponding to the graph of displacement versus time given in
[link] . (b) Identify the time or times (
,
,
, etc.) at which the instantaneous velocity is greatest. (c) At which times is it zero? (d) At which times is it negative?
(a) Explain how you can determine the acceleration over time from a velocity versus time graph such as the one in
[link] . (b) Based on the graph, how does acceleration change over time?
(a) Sketch a graph of acceleration versus time corresponding to the graph of velocity versus time given in
[link] . (b) Identify the time or times (
,
,
, etc.) at which the acceleration is greatest. (c) At which times is it zero? (d) At which times is it negative?
Consider the velocity vs. time graph of a person in an elevator shown in
[link] . Suppose the elevator is initially at rest. It then accelerates for 3 seconds, maintains that velocity for 15 seconds, then decelerates for 5 seconds until it stops. The acceleration for the entire trip is not constant so we cannot use the equations of motion from
Motion Equations for Constant Acceleration in One Dimension for the complete trip. (We could, however, use them in the three individual sections where acceleration is a constant.) Sketch graphs of (a) position vs. time and (b) acceleration vs. time for this trip.
A cylinder is given a push and then rolls up an inclined plane. If the origin is the starting point, sketch the position, velocity, and acceleration of the cylinder vs. time as it goes up and then down the plane.
Note: There is always uncertainty in numbers taken from graphs. If your answers differ from expected values, examine them to see if they are within data extraction uncertainties estimated by you.
(a) By taking the slope of the curve in
[link] , verify that the velocity of the jet car is 115 m/s at
. (b) By taking the slope of the curve at any point in
[link] , verify that the jet car’s acceleration is
.
Using approximate values, calculate the slope of the curve in
[link] to verify that the velocity at
is 0.208 m/s. Assume all values are known to 3 significant figures.
Using approximate values, calculate the slope of the curve in
[link] to verify that the velocity at
is 0.238 m/s. Assume all values are known to 3 significant figures.
Construct the displacement graph for the subway shuttle train as shown in
[link] (a). Your graph should show the position of the train, in kilometers, from t = 0 to 20 s. You will need to use the information on acceleration and velocity given in the examples for this figure.
(a) Take the slope of the curve in
[link] to find the jogger’s velocity at
. (b) Repeat at 7.5 s. These values must be consistent with the graph in
[link] .
A graph of
is shown for a world-class track sprinter in a 100-m race. (See
[link] ). (a) What is his average velocity for the first 4 s? (b) What is his instantaneous velocity at
? (c) What is his average acceleration between 0 and 4 s? (d) What is his time for the race?
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
from theory: distance [miles] = speed [mph] × time [hours]
info #1
speed_Dennis × 1.5 = speed_Wayne × 2
=> speed_Wayne = 0.75 × speed_Dennis (i)
info #2
speed_Dennis = speed_Wayne + 7 [mph] (ii)
use (i) in (ii) => [...]
speed_Dennis = 28 mph
speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5.
Substituting the first equation into the second:
W * 2 = (W + 7) * 1.5
W * 2 = W * 1.5 + 7 * 1.5
0.5 * W = 7 * 1.5
W = 7 * 3 or 21
W is 21
D = W + 7
D = 21 + 7
D = 28
Salma
Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67.
Check:
Sales = 3542
Commission 12%=425.04
Pay = 500 + 425.04 = 925.04.
925.04 > 925.00
Munster
difference between rational and irrational numbers
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?