A 0.6 kg block of ice is sliding by you on a very slippery floor at 2.5 m/s. As it goes by, you give it a kick perpendicular to its path. Your foot is in contact with the ice block for 0.0035 seconds. The block eventually slides at an angle of 24 degrees from its original direction (labeled

θ
in the diagram). The overhead view shown in the diagram is approximately to scale. The arrow represents the average force your toe applies briefly to the block of ice.

What is the unit vector in the direction of the block's momentum after the kick?

find the z-component of the block's momentum after the kick

To find the unit vector in the direction of the block's momentum after the kick, we first need to determine the momentum of the block after the kick.

Momentum (p) can be calculated using the formula:

p = m * v

Where:
p = momentum
m = mass of the block
v = velocity of the block

Given:
Mass of the block (m) = 0.6 kg
Velocity of the block before the kick (v) = 2.5 m/s

So, the initial momentum of the block can be calculated as:

p_initial = m * v = 0.6 kg * 2.5 m/s = 1.5 kg·m/s

Since the force is applied perpendicular to the direction of motion, it does not change the magnitude of the momentum. Therefore, the magnitude of the momentum after the kick remains the same.

Now, let's find the z-component of the block's momentum after the kick.

In the overhead view, the angle between the original direction of motion and the final direction (θ) is given as 24 degrees. The z-component is the component of momentum in the vertical direction.

We can use trigonometry to find the z-component of the momentum. Since the momentum after the kick has the same magnitude as the initial momentum, and the angle between the two directions is 24 degrees, we can write:

z-component of momentum = p_initial * sin(θ)

Substituting the given values:

z-component of momentum = 1.5 kg·m/s * sin(24 degrees)

Using a scientific calculator, we can evaluate sin(24 degrees) ≈ 0.4077.

z-component of momentum ≈ 1.5 kg·m/s * 0.4077

z-component of momentum ≈ 0.6116 kg·m/s

Therefore, the z-component of the block's momentum after the kick is approximately 0.6116 kg·m/s.

To find the unit vector in the direction of the block's momentum after the kick, we need to calculate the momentum vector.

Momentum (p) is defined as the product of mass (m) and velocity (v):

p = m * v

Given that the mass of the block is 0.6 kg and the velocity after the kick is at an angle of 24 degrees from its original direction, we can calculate the x-component (px) and y-component (py) of the momentum.

px = m * v * cos(24 degrees)
py = m * v * sin(24 degrees)

Next, we need to find the magnitude of the momentum vector using the Pythagorean theorem:

|p| = sqrt(px^2 + py^2)

Finally, to find the unit vector in the direction of the momentum, we divide the momentum vector by its magnitude:

p_unit_vector = p / |p|

Now, let's calculate the unit vector in the direction of the block's momentum after the kick.

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