A lighter block (2 kg) and a heavier block (8 kg) sit on a frictionless surface. Both blocks are initially at rest. The same force of 4 N then pushes to the right on each block for a distance of 9 m. What are the changes in kinetic energy of the blocks? What are the final kinetic energies of the blocks?What are the final momenta of the blocks?What are the changes in momenta of the blocks? How long does it take each block to travel the distance of 9 m?
a) W = Fd That's the change in KE
b) Same thing
c) Now you must find the final velocities. KE = 1/2mv^2, solve for v
momentum = mv
d) momentum is always conserved
e) a = F/m, v = at, solve for t
To solve this problem, we will use the principles of work, energy, and momentum.
1. Changes in kinetic energy of the blocks:
The change in kinetic energy of an object can be calculated using the formula: ΔKE = (1/2) * m * (vf^2 - vi^2), where ΔKE is the change in kinetic energy, m is the mass of the object, vf is the final velocity, and vi is the initial velocity.
Since both blocks are initially at rest, the initial velocities (vi) for both blocks are zero. To calculate the changes in kinetic energy, we can use the formula:
ΔKE1 = (1/2) * 2 kg * (vf1^2 - 0^2)
= (1/2) * 2 kg * vf1^2
= 1 kg * vf1^2
ΔKE2 = (1/2) * 8 kg * (vf2^2 - 0^2)
= (1/2) * 8 kg * vf2^2
= 4 kg * vf2^2
2. Final kinetic energies of the blocks:
The final kinetic energy of an object can be calculated using the formula: KE = (1/2) * m * vf^2, where KE is the kinetic energy, m is the mass of the object, and vf is the final velocity.
To calculate the final kinetic energies, we can use the formula:
KE1 = (1/2) * 2 kg * vf1^2
= 1 kg * vf1^2
KE2 = (1/2) * 8 kg * vf2^2
= 4 kg * vf2^2
3. Final momenta of the blocks:
The momentum of an object can be calculated using the formula: p = m * v, where p is the momentum, m is the mass of the object, and v is the velocity.
To calculate the final momenta, we can multiply the mass of each block by its final velocity:
p1 = 2 kg * vf1
p2 = 8 kg * vf2
4. Changes in momenta of the blocks:
The change in momentum of an object can be calculated using the formula: Δp = p - pi, where Δp is the change in momentum, p is the final momentum, and pi is the initial momentum.
Since both blocks are initially at rest, their initial momenta (pi) are zero. Therefore, the change in momentum is equal to the final momentum:
Δp1 = p1 - 0 = p1
Δp2 = p2 - 0 = p2
5. Time taken to travel the distance of 9 m:
The time taken can be calculated using the equation: distance = velocity * time.
Since both blocks experience the same force of 4 N, they will accelerate at the same rate. We can find this acceleration using Newton's second law.
F = ma
4 N = (2 kg + 8 kg) * a
4 N = 10 kg * a
a = 4 N / 10 kg
a = 0.4 m/s^2
To find the time taken, we use the equation:
distance = (1/2) * a * t^2
9 m = (1/2) * 0.4 m/s^2 * t^2
Simplifying:
9 m = 0.2 m/s^2 * t^2
t^2 = 45 s^2 / (m * m)
t ≈ 3.36 s
Now, let's calculate the values:
ΔKE1 = 1 kg * vf1^2
ΔKE2 = 4 kg * vf2^2
KE1 = 1 kg * vf1^2
KE2 = 4 kg * vf2^2
p1 = 2 kg * vf1
p2 = 8 kg * vf2
Δp1 = p1
Δp2 = p2
t ≈ 3.36 s
To find the changes in kinetic energy, final kinetic energies, final momenta, changes in momenta, and the time taken for each block to travel the distance of 9 m, we can use some basic concepts of physics.
1. Changes in Kinetic Energy:
The change in kinetic energy (∆KE) can be calculated using the equation:
∆KE = 0.5 * mass * (final velocity^2 - initial velocity^2)
For the lighter block (2 kg):
Initial velocity (vi) = 0 m/s (at rest)
Final velocity (vf) can be determined by using the equation:
force (F) = mass (m) * acceleration (a)
where acceleration (a) = force (F) / mass (m)
vf = acceleration * distance
For the heavier block (8 kg):
The same force of 4 N is applied, so the initial and final velocities can be calculated in the same way.
2. Final Kinetic Energies:
The final kinetic energy (KE) is given by:
KE = 0.5 * mass * velocity^2
Using the final velocity calculated earlier, we can find the final kinetic energy for each block.
3. Final Momenta:
The momentum (p) of an object is given by:
p = mass * velocity
Using the final velocity calculated earlier, we can find the final momentum for each block.
4. Changes in Momenta:
The change in momentum (∆p) can be calculated using the equation:
∆p = final momentum - initial momentum = final momentum - 0 (since initially at rest)
5. Time taken:
The time taken (t) to travel a distance (d) can be determined using the equation:
d = 0.5 * acceleration * t^2
Simplifying, we get:
t = sqrt(2 * d / acceleration)
Using the given information, we can plug in the values into the equations to find the answers to each question.