The justice league is playing baseball with 5kg rockets. The flash's top speed is 300m/s and a home run requires a 180m run. The rockets accerlate at 1200m/s^2, Superman catches it 30m from home, and decelerates it with 2700N. Once he stops the rocket, he applies a 2700N force for 2m throwing it 10m to thrid base. If he is safe Flash immediately begins to give high fives at a rate of 1 for every thousandth of a second, how many does he get to before the ball arrives? if he is out he has to buy superman a $2.99 hero hotdog for every hundredth of a second he was beaten, how much does he spend? What is flash's fate?
To find the answer to this question, we'll break it down into several steps:
Step 1: Calculate how long it takes for Superman to stop the rocket.
We can use Newton's second law of motion, F = ma, to find the acceleration of the rocket:
F = 2700N
m = 5kg
a = ?
Using the formula F = ma, we can rearrange the equation to solve for acceleration, a:
a = F/m
a = 2700N / 5kg
a = 540 m/s^2
Now, we can use the formula for acceleration, final velocity, and initial velocity to calculate the time it takes for the rocket to stop:
a = (v_f - v_i) / t
Rearranging the equation, we get:
t = (v_f - v_i) / a
v_f = 0m/s (final velocity, the rocket stops)
v_i = 300m/s (initial velocity, the Flash's top speed)
a = -540m/s^2 (negative because the rocket is decelerating)
Plugging in the values, we get:
t = (0 - 300) / -540
t = 0.556s (rounded to three decimal places)
Step 2: Calculate how long it takes for Superman to throw the rocket to third base.
Assuming Superman's force is applied uniformly over the 2m distance, we can calculate the acceleration using the formula for force:
F = ma
Rearranging the equation, we can solve for acceleration, a:
a = F/m
a = 2700N / 5kg
a = 540m/s^2
Now, we can use the formula for displacement, initial velocity, acceleration, and time to calculate the time it takes for Superman to throw the rocket to third base:
s = 10m (displacement)
u = 0m/s (initial velocity)
a = 540m/s^2
t = ?
The formula for displacement is:
s = ut + (1/2)at^2
Since initial velocity u is 0, the formula becomes:
s = (1/2)at^2
Plugging in the values, we get:
10 = (1/2)(540)t^2
20 = 540t^2
t^2 = 20/540
t ≈ 0.064s (rounded to three decimal places)
Step 3: Determine Flash's fate.
Flash's fate depends on whether he has enough time to reach home plate before the rocket arrives. To calculate this, we need to add the time it takes for Superman to stop the rocket (Step 1) and the time it takes for Superman to throw the rocket to third base (Step 2).
Total time = Step 1 + Step 2
Total time = 0.556s + 0.064s
Total time ≈ 0.62s
If Flash takes less than 0.62s to reach home plate, he is safe. Otherwise, he is out.
Step 4: Calculate the number of high fives Flash gets (if safe).
According to the question, Flash gives high fives at a rate of 1 for every thousandth of a second. To find the number of high fives Flash gives, we need to know how many thousandths of a second pass before the ball arrives.
The time it takes for the ball to arrive is the same as the total time calculated in Step 3: 0.62s.
To find the number of thousandths of a second, we multiply the total time by 1000:
Number of thousandths = Total time * 1000
Number of thousandths = 0.62s * 1000
Number of thousandths = 620
Therefore, Flash gets to give 620 high fives.
Step 5: Calculate the cost if Flash is out.
If Flash is out, according to the question, he has to buy Superman a $2.99 hero hotdog for every hundredth of a second he was beaten.
We calculated earlier that the total time it takes for the ball to arrive is 0.62s.
To find the number of hundredths of a second, we multiply the total time by 100:
Number of hundredths = Total time * 100
Number of hundredths = 0.62s * 100
Number of hundredths = 62
Therefore, Flash would have to spend 62 * $2.99, which is $185.38 if he is out.
So, to summarize:
- If Flash takes less than 0.62s to reach home plate, he is safe and gets to give 620 high fives.
- If Flash takes longer than 0.62s to reach home plate, he is out and has to spend $185.38 on hero hotdogs.
We can conclude Flash's fate and the cost he would incur once we have the time it takes for Flash to run the 180m.