# On a very muddy football field, a 100 kg linebacker tackles an 75 kg halfback. Immediately before the collision, the linebacker is slipping with a velocity of 8.5 m/s north and the halfback is sliding with a velocity of 7.5 m/s east.

What is the magnitude of the velocity at which the two players move together immediately after the collision?
What is the direction of this velocity?

## 100*8.5 = 175* Vnorth

75*7.5 = 175*Veast

|V| = sqrt(Vnorth^2+Veast^2)

T is angle East of north
(note, compass angle, not math angle)
tan T = Veast/Vnorth

## Well, it sounds like these football players are really getting down and dirty! Let's calculate the magnitude and direction of the velocity at which they move together after the collision.

To solve this, we can use the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.

The momentum of an object is the product of its mass and velocity. So, let's calculate the initial momentum of the linebacker and the halfback:

Initial momentum of the linebacker = mass of the linebacker × velocity of the linebacker
= 100 kg × 8.5 m/s north = 850 kg·m/s north

Initial momentum of the halfback = mass of the halfback × velocity of the halfback
= 75 kg × 7.5 m/s east = 562.5 kg·m/s east

Since momentum is a vector quantity, we need to add these momenta together taking their directions into account. Since the linebacker is moving north and the halfback is moving east, we have to use vector addition.

To do this, we can calculate the horizontal and vertical components of each momentum. Then we add up the horizontal components and the vertical components separately.

Horizontal component of the linebacker's momentum = 0 (since it's moving north)
Vertical component of the linebacker's momentum = 850 kg·m/s

Horizontal component of the halfback's momentum = 562.5 kg·m/s
Vertical component of the halfback's momentum = 0 (since it's moving east)

Now, let's add up the horizontal components and the vertical components:

Total horizontal momentum = Horizontal component of the linebacker's momentum + Horizontal component of the halfback's momentum
= 0 + 562.5 kg·m/s = 562.5 kg·m/s

Total vertical momentum = Vertical component of the linebacker's momentum + Vertical component of the halfback's momentum
= 850 kg·m/s + 0 = 850 kg·m/s

Now, we can use these components to find the magnitude and direction of the resulting velocity using the Pythagorean theorem and trigonometry.

Magnitude of the resulting velocity = √(Total horizontal momentum^2 + Total vertical momentum^2)
= √((562.5 kg·m/s)^2 + (850 kg·m/s)^2)
= √(316406.25 kg^2·m^2/s^2 + 722500 kg^2·m^2/s^2)
= √1035906.25 kg^2·m^2/s^2
≈ 1017.7 kg·m/s (rounded to two decimal places)

The direction of the resulting velocity can be found using trigonometry. The angle can be calculated as:

θ = arctan(Total vertical momentum/Total horizontal momentum)
= arctan(850 kg·m/s / 562.5 kg·m/s)
≈ 56.84 degrees (rounded to two decimal places)

So, the magnitude of the velocity at which the two players move together immediately after the collision is approximately 1017.7 kg·m/s, and the direction is approximately 56.84 degrees from the horizontal. Hope that answer tackles the question for you!

## To determine the magnitude of the velocity at which the two players move together immediately after the collision, we can use the principle of conservation of momentum.

The momentum before the collision can be calculated by multiplying the mass of each player by their respective velocities:

Momentum of linebacker before collision = 100 kg * 8.5 m/s north
Momentum of halfback before collision = 75 kg * 7.5 m/s east

To find the total momentum, we need to consider the directions. Since one player is moving north and the other is moving east, we'll have to consider the momentum as a vector quantity and calculate the total momentum using vector addition.

Using Pythagoras' theorem, we can calculate the magnitude of the total momentum:

Magnitude of total momentum before collision = sqrt((momentum of linebacker before collision)^2 + (momentum of halfback before collision)^2)

Now, for the direction of the velocity at which the two players move together immediately after the collision, we can use the direction of the momentum vector. We'll use trigonometry to determine the angle between the resultant momentum vector and the positive x-axis.

Using the arctangent function, we can calculate the angle:

Angle = arctan((momentum of linebacker before collision) / (momentum of halfback before collision))

This angle will give us the direction of the velocity.

Please note that we assume the collision is an elastic collision, where there is no external force acting on the system during the collision, and the total momentum before the collision is equal to the total momentum after the collision.

## To find the magnitude of the velocity at which the two players move together immediately after the collision, we can use the principle of conservation of momentum. The total momentum before the collision will be equal to the total momentum after the collision.

Given:
Mass of the linebacker (m1) = 100 kg
Mass of the halfback (m2) = 75 kg
Velocity of the linebacker before the collision (v1) = 8.5 m/s north
Velocity of the halfback before the collision (v2) = 7.5 m/s east

Step 1: Convert the velocities into components.
The velocity of the linebacker can be split into two components: one in the north direction and one in the east direction. Since the linebacker is slipping north, the north component is 8.5 m/s and the east component is 0 m/s.
The velocity of the halfback can be split into two components: one in the north direction and one in the east direction. Since the halfback is sliding east, the east component is 7.5 m/s and the north component is 0 m/s.

Step 2: Calculate the total momentum before the collision.
The total momentum before the collision (p_before) can be found by summing the individual momenta. The momentum (p) is given by mass (m) multiplied by velocity (v).

For the linebacker:
p1_before = m1 * v1 = 100 kg * 8.5 m/s north = 850 kg·m/s north

For the halfback:
p2_before = m2 * v2 = 75 kg * 7.5 m/s east = 562.5 kg·m/s east

Step 3: Calculate the total momentum after the collision.
Since momentum is conserved, the total momentum after the collision (p_after) will be the same as the total momentum before the collision.

For the linebacker and halfback together after the collision:
p_after = p1_after + p2_after

Step 4: Calculate the magnitude and direction of the velocity after the collision.
Since the mass and momentum are conserved, we know that the total mass before the collision is equal to the total mass after the collision. Therefore, we can use the total momentum after the collision to calculate the velocity (v_after) of the two players moving together.

v_after = p_after / (m1 + m2)

The magnitude of the velocity after the collision can be found using the Pythagorean theorem:
v_magnitude = √(v_after_north^2 + v_after_east^2)

To find the direction, we can use trigonometry. The direction (θ) can be found as:
θ = arctan(v_after_north / v_after_east)

Now, we can substitute the given values and solve for the magnitude and direction of the velocity after the collision.