An asteroid is moving along a straight line. A force acts along the displacement of the asteroid and slows it down. The asteroid has a mass of 4.5× 104 kg, and the force causes its speed to change from 7000 to 5000m/s. (a) What is the work done by the force? (b) If the asteroid slows down over a distance of 1.4× 106 m determine the magnitude of the force.
I know how to do part a, I'm not sure about part b though
work done = increase in kinetic energy
(here we have a decrease so the work is negative. The force is opposite to the direction of motion)
work = (1/2)(4.5*10^4)(5000^2 - 7000^2)
work = force *distance moved in direction of force
so
Force = work/(1.4*10^6)
Well, in order to determine the magnitude of the force, we need to consider both the work done and the distance over which the force is applied.
First, let's tackle part (a). 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. So, the work done by the force can be calculated as:
Work = ΔKE = (1/2)mv²f - (1/2)mv²i
Substituting the given values, we have:
Work = (1/2)(4.5×10⁴)(5000²) - (1/2)(4.5×10⁴)(7000²)
Now, if you solve this equation you'll get the value of work done by the force.
Moving on to part (b), we can use the work-work rate relationship to calculate the magnitude of the force. The equation for work is:
Work = Force × Distance
Since the force is acting against the motion (slowing down the asteroid), it is in the opposite direction of the displacement. So we can rewrite it as:
Work = -Force × Distance
Rearranging the equation to solve for the magnitude of the force:
Force = -Work / Distance
Plug in the value of work from part (a) and the given distance to find the magnitude of the force.
Remember, even though we are using a bit of physics here, it doesn't mean we can't have some fun! So, while you crunch those numbers, here's a little joke for you:
Why did the asteroid go to therapy?
Because it had some serious space issues!
To solve part (b) of this problem, 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.
In this case, the work done on the asteroid is the amount of energy lost due to the force that is slowing it down. We can calculate this as follows:
Step 1: Calculate the initial kinetic energy (KEi) of the asteroid using the equation:
KEi = 0.5 * mass * (initial velocity)^2
KEi = 0.5 * 4.5×10^4 kg * (7000 m/s)^2
Step 2: Calculate the final kinetic energy (KEf) of the asteroid using the equation:
KEf = 0.5 * mass * (final velocity)^2
KEf = 0.5 * 4.5×10^4 kg * (5000 m/s)^2
Step 3: Calculate the change in kinetic energy (ΔKE) using the equation:
ΔKE = KEf - KEi
Step 4: The work done by the force (W) can be calculated as:
W = -ΔKE
Since the force is causing the object to slow down, the work done on it is negative.
Now, let's perform the calculations:
Step 1: KEi = 0.5 * 4.5×10^4 kg * (7000 m/s)^2
KEi = 0.5 * 4.5×10^4 kg * 49×10^6 m^2/s^2
KEi = 110.25×10^9 J
Step 2: KEf = 0.5 * 4.5×10^4 kg * (5000 m/s)^2
KEf = 0.5 * 4.5×10^4 kg * 25×10^6 m^2/s^2
KEf = 56.25×10^9 J
Step 3: ΔKE = KEf - KEi
ΔKE = 56.25×10^9 J - 110.25×10^9 J
ΔKE = -54×10^9 J
Step 4: W = -ΔKE
W = -(-54×10^9 J)
W = 54×10^9 J
The work done by the force is 54×10^9 J.
Now, let's move to part (b) of the question.
We can use the equation for work:
W = force * distance
The force in this case is the magnitude of the force that is slowing down the asteroid.
Given that initially the asteroid has a speed of 7000 m/s and that it slows down over a distance of 1.4×10^6 m, we can substitute these values into the equation:
54×10^9 J = force * 1.4×10^6 m
Solving for the force:
force = 54×10^9 J / 1.4×10^6 m
force ≈ 3.86×10^4 N
The magnitude of the force is approximately 3.86×10^4 N.
To determine the magnitude of the force, we can use the work-energy theorem, which states that the work done on an object equals the change in its kinetic energy. The work is given by the equation:
Work = Change in kinetic energy
In this case, the asteroid slows down, so we can find the work done by calculating the change in kinetic energy:
Change in kinetic energy = (1/2) * mass * (final velocity^2 - initial velocity^2)
Given:
Mass of the asteroid (m) = 4.5 × 10^4 kg
Initial velocity (v_i) = 7000 m/s
Final velocity (v_f) = 5000 m/s
Change in kinetic energy = (1/2) * m * (v_f^2 - v_i^2)
= (1/2) * (4.5 × 10^4 kg) * (5000 m/s)^2 - (7000 m/s)^2
Simplifying the equation:
Change in kinetic energy = (1/2) * (4.5 × 10^4 kg) * (25 × 10^6 m^2/s^2 - 49 × 10^6 m^2/s^2)
= (1/2) * (4.5 × 10^4 kg) * (-24 × 10^6 m^2/s^2)
Now, we can calculate the change in kinetic energy:
Change in kinetic energy = -10.8 × 10^10 J
Since work equals the change in kinetic energy, the work done by the force acting on the asteroid is:
Work = -10.8 × 10^10 J
For part b, we can use the definition of work to find the magnitude of the force. The work done by a force is given by the equation:
Work = Force * distance * cos(theta)
Where:
Force is the magnitude of the force applied
Distance is the distance over which the force is applied
Theta is the angle between the force and the displacement
In this case, the distance is given as 1.4 × 10^6 m. Since the force acts along the displacement, the angle between the force and the displacement is 0º, leading to cos(0º) = 1.
Therefore, we can rewrite the equation as:
Work = Force * distance
Since we've already calculated the work as -10.8 × 10^10 J, we can substitute the values and solve for the magnitude of the force:
-10.8 × 10^10 J = Force * (1.4 × 10^6 m)
Simplifying the equation:
Force * (1.4 × 10^6 m) = -10.8 × 10^10 J
Now, we can solve for the magnitude of the force (F):
Force = (-10.8 × 10^10 J) / (1.4 × 10^6 m)
Calculating the magnitude of the force:
Force = -77.14 N
Since magnitude cannot be negative, the magnitude of the force acting on the asteroid is 77.14 N.