A sprinter starts from rest and reaches his top speed in 4 seconds running with a constant acceleration. He then maintains that top speed and finishes the 100 m in the overall time of 10 seconds. What is his acceleration?
how would i know dummy
To find the acceleration of the sprinter, we can use the equation of motion:
v = u + at
where:
v = final velocity
u = initial velocity
a = acceleration
t = time taken
First, let's find the initial velocity (u) of the sprinter. Since the sprinter starts from rest (rest implies zero initial velocity), we have u = 0.
Next, let's find the final velocity (v) of the sprinter. We know that the sprinter reaches his top speed and then maintains that speed for the remaining distance. Therefore, the final velocity (v) is the top speed of the sprinter.
Now, let's find the final velocity (v) using the equation:
v = u + at
Since the sprinter starts from rest and reaches his top speed in 4 seconds (u = 0, t = 4 seconds), we have:
v = 0 + a * 4
v = 4a
Now, we know that the sprinter maintains this top speed for the remaining distance. The time taken for this phase is 10 seconds - 4 seconds = 6 seconds.
Using the equation v = u + at again, where u is the final velocity (v) obtained when reaching top speed, we have:
v = u + at
v = 4a + a * 6
v = 4a + 6a
v = 10a
We can also express the distance covered in terms of the initial velocity (u), acceleration (a), and time (t) using the equation:
s = ut + (1/2)at^2
Since the sprinter starts from rest (u = 0), the equation simplifies to:
s = (1/2)at^2
100 = (1/2) * a * 10^2
100 = 50a
Now, we have two equations:
v = 10a
100 = 50a
We can solve these equations simultaneously to find the value of acceleration (a):
10a = 100
a = 100 / 10
a = 10 m/s^2
Therefore, the acceleration of the sprinter is 10 m/s^2.
Total distance covered is the sum of the distance covered while accelerating (for t = 4 seconds) plus the distance travelled at constant speed (V = at = 4a) for 6 seconds. Put that into equation form.
100 m = (1/2)at^2 + (at)*6
= 8a + 24a
and solve for a