While pushing off against the starting blocks at the beginning of a race, Olympic swimming great

Michael Phelps generates a force that increases linearly from 0 N to 800 N over the first 0.30 s, and then
decreases steadily back to 0 N over the remaining 0.30 s (see the graph below showing the force versus time).
If his mass is 86 kg and he was initially at rest, what was his velocity at the instant he left the starting
blocks? (Hint: it is easiest to solve this problem using impulse-momentum.)

To find Michael Phelps' velocity at the instant he left the starting blocks, we can use the concept of impulse-momentum.

Impulse is defined as the change in momentum of an object, which can be calculated using the formula:

Impulse = Force x Time

In this case, the impulse will be equal to the change in momentum of Michael Phelps. The impulse can be calculated by finding the area under the force versus time graph.

From the graph, we can see that the force increases linearly from 0 N to 800 N over the first 0.30 s, reaching its maximum value. This means that the force remains constant at 800 N for the next 0.30 s before decreasing back to 0 N.

To find the impulse, we need to find the area under the force versus time graph. Since the graph consists of two parts - a triangle and a rectangle - we need to calculate the areas of each part separately.

Area of the triangle = (1/2) x base x height = (1/2) x 0.3 s x 800 N = 120 N⋅s

Area of the rectangle = length x width = 0.3 s x 800 N = 240 N⋅s

Total impulse = Area of the triangle + Area of the rectangle = 120 N⋅s + 240 N⋅s = 360 N⋅s

Now, we can use the impulse-momentum equation to find the change in momentum of Michael Phelps:

Impulse = Change in momentum

360 N⋅s = mass x change in velocity

Here, the mass of Michael Phelps is given as 86 kg, and he was initially at rest, so his initial velocity (u) is 0 m/s.

360 N⋅s = 86 kg x change in velocity

change in velocity = 360 N⋅s / 86 kg

Therefore, Michael Phelps' change in velocity is approximately 4.19 m/s.

Since his initial velocity was 0 m/s, his final velocity when he left the starting blocks is 4.19 m/s.

To find the velocity of Michael Phelps at the instant he left the starting blocks, we can use the principle of impulse-momentum. The impulse experienced by an object is equal to the change in its momentum.

The impulse is given by the product of force and time:

Impulse = Force × Time

In the first 0.30 seconds, the force increases linearly from 0 N to 800 N. So, we can find the impulse during this time interval:

Impulse = (Average Force) × (Time Interval)

The average force can be calculated by taking the average of the initial and final forces:

Average Force = (Initial Force + Final Force) / 2

Next, we can calculate the time interval:

Time Interval = Final Time - Initial Time

Since we know the mass of Michael Phelps is 86 kg and he starts from rest, we can use the formula for impulse:

Impulse = Change in Momentum = Mass × Change in Velocity

Now, let's break down the calculation step by step:

Step 1: Calculate the average force during the first 0.30 seconds.
Average Force = (0 N + 800 N) / 2 = 400 N

Step 2: Calculate the time interval for the first 0.30 seconds.
Time Interval = 0.30 s - 0 s = 0.30 s

Step 3: Calculate the impulse during the first 0.30 seconds.
Impulse = Average Force × Time Interval = 400 N × 0.30 s = 120 Ns

Step 4: Calculate the change in velocity using the impulse-momentum formula.
Impulse = Mass × Change in Velocity
120 Ns = 86 kg × (Final Velocity - 0 m/s)

Simplifying the equation:

Final Velocity = 120 Ns / 86 kg

Final Velocity ≈ 1.395 m/s

Therefore, the velocity of Michael Phelps at the instant he left the starting blocks is approximately 1.395 m/s.