In a relay race the second runner does not start from rest. it covers 100 m in 10 sec, finishing with maximum velocity of 12 m/sec, assuming the constant acceleration, determine his initial velocity. ??
To determine the initial velocity of the second runner, we can use the equation of motion:
v^2 = u^2 + 2as
Where:
- v is the final velocity (12 m/s)
- u is the initial velocity (unknown)
- a is the acceleration (unknown)
- s is the distance covered (100 m)
Since the acceleration is assumed to be constant, we can rearrange the equation to solve for the initial velocity:
u^2 = v^2 - 2as
Substituting the given values, we have:
u^2 = (12^2) - 2(100)(a)
Simplifying further:
u^2 = 144 - 200a
We also know that the time taken to cover the distance can be calculated using the equation:
v = u + at
Since the second runner covers the distance of 100 m in 10 seconds, we have:
12 = u + (10)(a)
Now we have a system of equations:
u^2 = 144 - 200a (Equation 1)
12 = u + 10a (Equation 2)
To solve this system, let's solve Equation 2 for u and substitute it into Equation 1:
u = 12 - 10a
Substituting into Equation 1:
(12 - 10a)^2 = 144 - 200a
Expanding and simplifying:
144 - 240a + 100a^2 = 144 - 200a
Rearranging and combining like terms:
100a^2 - 40a = 0
Factoring out common terms:
a(100a - 40) = 0
Setting each factor to zero and solving for a:
a = 0 or a = 0.4
Since the runner is accelerating, the acceleration cannot be zero. Therefore, a = 0.4 m/s^2.
Now, substituting this value of a into Equation 2:
12 = u + 10(0.4)
12 = u + 4
Rearranging and solving for u:
u = 12 - 4
u = 8 m/s
Therefore, the initial velocity of the second runner is 8 m/s.
To determine the initial velocity of the second runner in the relay race, we can use the equations of motion for constant acceleration.
Here's how you can solve it step-by-step:
Step 1: Identify the known values from the problem:
- Distance covered by the runner (s) = 100 m
- Time taken (t) = 10 sec
- Maximum velocity (v) = 12 m/sec
Step 2: Recall the equation of motion for constant acceleration:
v = u + at
In this equation:
- v is the final velocity
- u is the initial velocity
- a is the acceleration
- t is the time
Step 3: Rearrange the equation to solve for the initial velocity (u):
u = v - at
Step 4: Plug in the known values into the equation:
u = 12 m/sec - a × 10 sec
Step 5: We can determine the acceleration by using the formula for average acceleration:
Average acceleration (a) = (change in velocity) / (time taken)
Since the runner starts from rest (initial velocity is 0 m/sec) and reaches a maximum velocity of 12 m/sec, the change in velocity is 12 m/sec.
a = (change in velocity) / (time taken)
= (12 m/sec - 0 m/sec) / 10 sec
= 12 m/sec / 10 sec
= 1.2 m/sec²
Step 6: Substitute the acceleration back into the equation:
u = 12 m/sec - 1.2 m/sec² × 10 sec
Step 7: Solve for the initial velocity:
u = 12 m/sec - 12 m/sec
u = 0 m/sec
Therefore, the initial velocity of the second runner is 0 m/sec.