Use principles of force and kinematics to answer these:
a.
A 40 kg skater pushes on the 60 kg skater (initially at rest). As they collide, the 40 kg skater pushes on the 60 kg skater, applying a force 100 Newtons for 1.2 seconds. Find the velocity at which each of them ends up moving (they should be equal).
b.
A 60kg skater slides at 3.0 m/s toward a 40 kg skater (initially at rest). They collide and hold on, so both have the same final speed. find that speed. You don't need to know the Force and the time, it will work out.
a. To find the velocity at which each skater ends up moving, we can use the principles of force and kinematics. Since the collision is happening in one dimension, we can assume it is along a straight line and use the principle of conservation of momentum.
The momentum before the collision is given by the equation:
(m1 * v1) + (m2 * v2) = (m1 * u1) + (m2 * u2)
Where:
m1 = mass of the 40 kg skater = 40 kg
v1 = final velocity of the 40 kg skater (unknown)
m2 = mass of the 60 kg skater = 60 kg
v2 = final velocity of the 60 kg skater (unknown)
u1 = initial velocity of the 40 kg skater = 0 m/s (initially at rest)
u2 = initial velocity of the 60 kg skater = 0 m/s (initially at rest)
Using the given information, the equation becomes:
(40 kg * v1) + (60 kg * v2) = (40 kg * 0 m/s) + (60 kg * 0 m/s)
Simplifying the equation, we have:
40 kg * v1 + 60 kg * v2 = 0 kg * m/s
Next, we can use the principle of force and kinematics to relate the force applied to the change in momentum:
Force = (change in momentum) / (time)
The change in momentum is given by:
(change in momentum) = (final momentum) - (initial momentum)
Since the initial momentum is zero (both skaters are initially at rest), the equation becomes:
(change in momentum) = (final momentum)
The final momentum is given by:
(final momentum) = (m1 * v1) + (m2 * v2)
Using the given force of 100 Newtons and time of 1.2 seconds, we can write:
100 N = [(m1 * v1) + (m2 * v2)] / 1.2 s
Substituting the values of m1 = 40 kg and m2 = 60 kg, we have:
100 N = [(40 kg * v1) + (60 kg * v2)] / 1.2 s
Rearranging the equation, we get:
4000 N * s = (40 kg * v1) + (60 kg * v2)
Since we have two unknowns (v1 and v2), we need another equation to solve for them. One way to do this is by considering the conservation of kinetic energy. Assuming there are no external forces acting on the skaters, the total kinetic energy before and after the collision should be the same.
The initial kinetic energy is given by:
(initial kinetic energy) = (0.5 * m1 * u1^2) + (0.5 * m2 * u2^2)
Since both skaters are initially at rest (u1 = u2 = 0 m/s), the initial kinetic energy is 0.
The final kinetic energy is given by:
(final kinetic energy) = (0.5 * m1 * v1^2) + (0.5 * m2 * v2^2)
Setting the initial and final kinetic energies equal to each other, we have:
0 = (0.5 * 40 kg * v1^2) + (0.5 * 60 kg * v2^2)
Simplifying the equation, we get:
0 = 20 kg * v1^2 + 30 kg * v2^2
Now we have two equations:
40 kg * v1 + 60 kg * v2 = 4000 N * s
20 kg * v1^2 + 30 kg * v2^2 = 0
We can solve these equations simultaneously to find the values of v1 and v2.
b. In this case, both skaters collide and hold on, so their final speed will be the same. To find that speed, we need to consider the conservation of momentum.
Using the principle of conservation of momentum, we can write:
(m1 * v1) + (m2 * v2) = (m1 + m2) * v_final
Where:
m1 = mass of the 60 kg skater = 60 kg
v1 = initial velocity of the 60 kg skater = 3.0 m/s
m2 = mass of the 40 kg skater = 40 kg
v2 = initial velocity of the 40 kg skater = 0 m/s (initially at rest)
v_final = final velocity of both skaters (unknown)
Substituting the known values, the equation becomes:
(60 kg * 3.0 m/s) + (40 kg * 0 m/s) = (60 kg + 40 kg) * v_final
Simplifying the equation, we have:
180 kg * m/s = 100 kg * v_final
Dividing both sides of the equation by 100 kg, we get:
1.8 m/s = v_final
Therefore, the final speed at which both skaters end up moving is 1.8 m/s.
a. To find the velocity at which each skater ends up moving, we can use the principle of force and kinematics. First, we need to calculate the initial acceleration of the skaters.
Using Newton's second law of motion, F = ma, where F is the force applied and m is the mass of the skater, we can determine the acceleration:
For the 40 kg skater:
F = m * a
100 N = 40 kg * a
a = 100 N / 40 kg
a = 2.5 m/s^2
Now that we have the acceleration, we can use the kinematic equation to find the final velocity of each skater:
vf = vi + at
Since both skaters started at rest (vi = 0 m/s), we can simplify the equation to:
vf = at
For the 40 kg skater:
vf1 = 2.5 m/s^2 * 1.2 s
vf1 = 3 m/s
For the 60 kg skater (since the skaters end up moving at the same velocity):
vf2 = vf1 = 3 m/s
Therefore, both skaters end up moving at a velocity of 3 m/s.
b. In this case, since both skaters have the same final speed after the collision, we can solve it using the conservation of momentum principle.
Using the equation for conservation of momentum:
(m1 * v1) + (m2 * v2) = (m1 + m2) * vf
where m1 and m2 are the masses of the skaters, v1 and v2 are their initial velocities, and vf is their final velocity.
Since the first skater is moving at 3.0 m/s and the second skater is initially at rest, we have:
(60 kg * 3.0 m/s) + (40 kg * 0 m/s) = (60 kg + 40 kg) * vf
180 kg m/s = 100 kg * vf
Dividing both sides by 100 kg:
1.8 m/s = vf
Therefore, the final speed at which both skaters move after the collision is 1.8 m/s.