What else can we learn by examining the equation
We see that:
displacement depends on the square of the elapsed time when acceleration is not zero. In
[link] , the dragster covers only one fourth of the total distance in the first half of the elapsed time
if acceleration is zero, then the initial velocity equals average velocity (
) and
becomes
Solving for final velocity when velocity is not constant (
)
A fourth useful equation can be obtained from another algebraic manipulation of previous equations.
If we solve
for
, we get
Substituting this and
into
, we get
Calculating final velocity: dragsters
Calculate the final velocity of the dragster in
[link] without using information about time.
Strategy
Draw a sketch.
The equation
is ideally suited to this task because it relates velocities, acceleration, and displacement, and no time information is required.
Solution
1. Identify the known values. We know that
, since the dragster starts from rest. Then we note that
(this was the answer in
[link] ). Finally, the average acceleration was given to be
.
2. Plug the knowns into the equation
and solve for
Thus
To get
, we take the square root:
Discussion
145 m/s is about 522 km/h or about 324 mi/h, but even this breakneck speed is short of the record for the quarter mile. Also, note that a square root has two values; we took the positive value to indicate a velocity in the same direction as the acceleration.
An examination of the equation
can produce further insights into the general relationships among physical quantities:
The final velocity depends on how large the acceleration is and the distance over which it acts
For a fixed deceleration, a car that is going twice as fast doesn't simply stop in twice the distance—it takes much further to stop. (This is why we have reduced speed zones near schools.)
Putting equations together
In the following examples, we further explore one-dimensional motion, but in situations requiring slightly more algebraic manipulation. The examples also give insight into problem-solving techniques. The box below provides easy reference to the equations needed.
Summary of kinematic equations (constant
)
Calculating displacement: how far does a car go when coming to a halt?
On dry concrete, a car can decelerate at a rate of
, whereas on wet concrete it can decelerate at only
. Find the distances necessary to stop a car moving at 30.0 m/s
(about 110 km/h) (a) on dry concrete and (b) on wet concrete. (c) Repeat both calculations, finding the displacement from the point where the driver sees a traffic light turn red, taking into account his reaction time of 0.500 s to get his foot on the brake.
Strategy
Draw a sketch.
In order to determine which equations are best to use, we need to list all of the known values and identify exactly what we need to solve for. We shall do this explicitly in the next several examples, using tables to set them off.
Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ...
Step 2: Find each score's deviation from the mean. ...
Step 3: Square each deviation from the mean. ...
Step 4: Find the sum of squares. ...
Step 5: Divide the sum of squares by n – 1 or N.
The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.
Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400
a. what is the probability of getting more than 12,000 hits?
b. what is the probability of getting fewer than 9,000 hits?
Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400.
a. What is the probability of getting more than 12,000 hits