A 32.0kg crate is initially moving with a velocity that has magnitude 3.92m/s in a direction 37.0 ∘ west of north.How much work must be done on the crate to change its velocity to 5.50m/s in a direction 63.0 ∘ south of east?
You are given an initial KEnergy, a final KE, and looking for the added energy.
initial KE + added work=finalKE
Notice it does not depend on direction, or time.
1/2 m 3.92^2+work=1/2 m 5.5^2
solve for work. mass m is given.
238
Well, well, well, looks like this crate is going on a wild ride! Let's calculate the work that needs to be done, shall we?
First, let's break down the initial velocity vector. The magnitude is 3.92 m/s, and the direction is 37.0° west of north. So, the initial velocity in the north direction is 3.92 × cos(37.0°) and the initial velocity in the west direction is 3.92 × sin(37.0°).
To calculate the work done to change the velocity, we need to figure out the change in velocity. The final velocity has a magnitude of 5.50 m/s and a direction 63.0° south of east. So, the final velocity in the east direction is 5.50 × cos(63.0°) and the final velocity in the south direction is 5.50 × sin(63.0°).
Now, let's put on our calculating hats and crunch some numbers. The change in velocity in the east direction is the final velocity in the east direction minus the initial velocity in the east direction. Similarly, the change in velocity in the south direction is the final velocity in the south direction minus the initial velocity in the south direction.
Finally, the work done on the crate is given by the equation: work = mass × change in velocity.
So, plug in the values, do the math, and voilà! You'll have the work done on the crate. Go on, give it a try!
To find the work done on the crate, we need to calculate the change in kinetic energy using the work-energy theorem. The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy.
1. Calculate the initial kinetic energy:
The initial kinetic energy (KEi) is given by the formula:
KEi = (1/2) * mass * velocity^2
Substituting the given values:
mass = 32.0 kg
velocity = 3.92 m/s
Thus, KEi = (1/2) * 32.0 kg * (3.92 m/s)^2
2. Calculate the final kinetic energy:
The final kinetic energy (KEf) is given by the formula:
KEf = (1/2) * mass * velocity^2
Substituting the given values:
mass = 32.0 kg
velocity = 5.50 m/s
Thus, KEf = (1/2) * 32.0 kg * (5.50 m/s)^2
3. Calculate the change in kinetic energy:
The change in kinetic energy (ΔKE) is given by the formula:
ΔKE = KEf - KEi
Subtracting the initial kinetic energy from the final kinetic energy:
ΔKE = [(1/2) * 32.0 kg * (5.50 m/s)^2] - [(1/2) * 32.0 kg * (3.92 m/s)^2]
4. Calculate the work done:
The work done (W) is equal to the change in kinetic energy (ΔKE).
Substituting the calculated value of ΔKE:
W = ΔKE
Thus, the work done on the crate to change its velocity is equal to the change in kinetic energy, which is calculated in step 3.
To find the work done on the crate, 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.
First, let's find the initial and final kinetic energies of the crate. The kinetic energy (KE) of an object is given by the equation KE = (1/2)mv^2, where m is the mass of the object and v is its velocity.
Given:
Mass of the crate (m) = 32.0 kg
Initial velocity (v) = 3.92 m/s
Final velocity (v') = 5.50 m/s
1. Initial Kinetic Energy (KE1):
KE1 = (1/2)mv^2
= (1/2)(32.0 kg)(3.92 m/s)^2
2. Final Kinetic Energy (KE2):
KE2 = (1/2)mv'^2
= (1/2)(32.0 kg)(5.50 m/s)^2
Next, we need to find the work done (W) on the crate, which is equal to the change in kinetic energy (ΔKE).
ΔKE = KE2 - KE1
To calculate ΔKE, we subtract the initial kinetic energy from the final kinetic energy.
ΔKE = KE2 - KE1
Now we can substitute the values into the equation:
ΔKE = (1/2)(32.0 kg)(5.50 m/s)^2 - (1/2)(32.0 kg)(3.92 m/s)^2
Simplifying the equation, we get:
ΔKE ≈ (1/2)(32.0 kg)(30.25 m^2/s^2) - (1/2)(32.0 kg)(15.3664 m^2/s^2)
ΔKE ≈ 487.52 J - 246.09 J
Finally, we subtract the initial kinetic energy from the final kinetic energy to determine the change in kinetic energy:
ΔKE ≈ 241.43 J
Therefore, the work done on the crate to change its velocity is approximately 241.43 Joules.