A sprinter in a race can run 100m in 11.4 seconds. If it takes him 4.2 seconds to accelerate to top speed, what is the magnitude of his acceleration? Assume that he stays at his top speed from 4.2 seconds till the end. I do not understand which kinematics equation will help, as they all have multiple variables.
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To solve this problem, you can use the kinematic equation that relates displacement, initial velocity, final velocity, time, and acceleration. The equation is:
d = (v0 + v)t/2
where d is the displacement, v0 is the initial velocity, v is the final velocity, t is the time, and a is the acceleration.
In this problem, we need to find the magnitude of the acceleration. We are given the total time to run 100m, which is 11.4 seconds, and the time it takes to accelerate to top speed, which is 4.2 seconds. We also know that the sprinter stays at his top speed until the end.
Let's break down the problem into two parts: the acceleration phase and the constant top speed phase.
1. Acceleration Phase:
During the first 4.2 seconds of the race, the sprinter is accelerating to reach his top speed. We are given the time and the distance covered during this phase (0-4.2 seconds). The equation we can use to find the acceleration during this phase is:
d = (v0 + v)t/2
We know that the initial velocity, v0, is 0, as the sprinter starts from rest. The final velocity, v, is the top speed he reaches after 4.2 seconds. We also know the time, t, which is 4.2 seconds, and the distance, d, which is the total distance covered during the acceleration phase, which is unknown.
2. Constant Top Speed Phase:
After the acceleration phase, the sprinter stays at his top speed until the end of the race. We know that the total time for the race is 11.4 seconds, and the time taken for the acceleration phase is 4.2 seconds. Therefore, the time for the constant top speed phase can be calculated as:
t2 = total time - acceleration time = 11.4 s - 4.2 s = 7.2 s
In this phase, the sprinter maintains the same speed for the entire duration of 7.2 seconds. We can find the distance covered during this phase by multiplying the average speed (which is the top speed) by the time:
d2 = v × t2
Now, to find the magnitude of the acceleration, we need to calculate the final velocity, v, which is the top speed. We can do this by finding the distance covered during the acceleration phase, d, and the time, t, using the equation:
d = (v0 + v)t/2
Since v0 is 0, the equation simplifies to:
d = vt/2
Rearranging the equation, we get:
v = 2d/t
By substituting the given values, we can calculate the final velocity, v.
Finally, to find the magnitude of the acceleration, we can use the following equation:
a = (v - v0)/t
Since v0 is 0, the equation simplifies to:
a = v/t
By substituting the values of v and t, we can calculate the magnitude of the acceleration.