A 5.0 104 kg space probe is traveling at a speed of 11000 m/s through deep space. Retrorockets are fired along the line of motion to reduce the probe's speed. The retrorockets generate a force of 4.5 105 N over a distance of 2600 km. What is the final speed of the probe?

calculate initial Kinetic energy

Ke = (1/2) m v^2 Joules

calculate work done by force
W = 4.5 *10^5 * 2600 * 10^3 Joules

New Ke = original Ke - W

1/2) m Vfinal^2 = New Ke

I continue to plug in 1.855 x 10^12 as the answer into my online hw and it rejected the answer. Is that correct?

To find the final speed of the probe, we can use the principle of work and energy.

The work done on an object is equal to the force applied to it multiplied by the distance over which the force is applied. In this case, the work done on the probe by the retrorockets is given by:

Work = Force * Distance

W = 4.5 * 10^5 N * 2.6 * 10^6 m (converting 2600 km to meters)

Next, we can use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy. The initial kinetic energy of the probe is given by:

Initial Kinetic Energy = 0.5 * mass * (initial speed)^2

KEi = 0.5 * 5.0 * 10^4 kg * (11000 m/s)^2

The final kinetic energy of the probe is given by:

Final Kinetic Energy = 0.5 * mass * (final speed)^2

KEf = 0.5 * 5.0 * 10^4 kg * (final speed)^2

According to the work-energy principle, the change in kinetic energy is given by:

Change in Kinetic Energy = Final Kinetic Energy - Initial Kinetic Energy

ΔKE = KEf - KEi

Since the work done by the retrorockets is equal to the change in kinetic energy, we have:

W = ΔKE

4.5 * 10^5 N * 2.6 * 10^6 m = (0.5 * 5.0 * 10^4 kg * (final speed)^2) - (0.5 * 5.0 * 10^4 kg * (11000 m/s)^2)

Simplifying the equation and solving for the final speed:

(4.5 * 10^5 N * 2.6 * 10^6 m) / (0.5 * 5.0 * 10^4 kg) = (final speed)^2 - (11000 m/s)^2

(4.5 * 10^5 N * 2.6 * 10^6 m) / (0.5 * 5.0 * 10^4 kg) + (11000 m/s)^2 = (final speed)^2

11000 m/s + sqrt[(4.5 * 10^5 N * 2.6 * 10^6 m) / (0.5 * 5.0 * 10^4 kg) + (11000 m/s)^2] = final speed

Calculating the final speed:

final speed = 11000 m/s + sqrt[(4.5 * 10^5 N * 2.6 * 10^6 m) / (0.5 * 5.0 * 10^4 kg) + (11000 m/s)^2]

final speed ≈ 11000 m/s + sqrt[0.23984 + 121000000]

final speed ≈ 11000 m/s + sqrt[121000000.23984]

final speed ≈ 11000 m/s + 11000.1575

final speed ≈ 22100.1575 m/s

Therefore, the final speed of the probe is approximately 22,100.16 m/s (rounded to two decimal places).

To find the final speed of the probe, we can use the concept of work and energy.

First, let's calculate the work done by the retrorockets. The work done is equal to the force multiplied by the distance over which the force is applied.

Given:
Force (F) = 4.5 * 10^5 N
Distance (d) = 2600 km = 2.6 * 10^6 m

Using the formula for work (W) = F * d, we can calculate the work done:

W = (4.5 * 10^5 N) * (2.6 * 10^6 m)
W = 11.7 * 10^11 Nm

Next, we can use the work-energy principle. According to this principle, the work done on an object is equal to the change in its kinetic energy.

The initial kinetic energy of the probe is given by:
Initial Kinetic Energy = (1/2) * mass * (initial velocity)^2

Given:
Mass (m) = 5.0 * 10^4 kg
Initial velocity (v) = 11000 m/s

Initial Kinetic Energy = (1/2) * (5.0 * 10^4 kg) * (11000 m/s)^2

To simplify the calculation, let's first find the velocity squared:

(11000 m/s)^2 = 121000000 m^2/s^2

Now, we can calculate the initial kinetic energy:

Initial Kinetic Energy = (1/2) * (5.0 * 10^4 kg) * (121000000 m^2/s^2)
Initial Kinetic Energy = 3.025 * 10^14 J

According to the work-energy principle, the change in kinetic energy is equal to the work done:

Change in Kinetic Energy = Work done

Since the retrorockets are applied to reduce the speed, the change in kinetic energy is negative. Thus:

- (Change in Kinetic Energy) = 11.7 * 10^11 Nm

Finally, we can find the final kinetic energy using the formula:

Final Kinetic Energy = Initial Kinetic Energy - (Change in Kinetic Energy)

Final Kinetic Energy = 3.025 * 10^14 J - (11.7 * 10^11 Nm)

Now, to find the final speed, we can use the formula for kinetic energy:

Final Kinetic Energy = (1/2) * mass * (final velocity)^2

We can rearrange this formula to solve for the final velocity:

(final velocity)^2 = (2 * Final Kinetic Energy) / mass

Let's substitute the known values into the equation and solve for the final velocity.