skater n 58.9 kg is moving north at a speed of 7.8 m/s when she collides with skate E 72.6 kg moving east at 3.5 m/s the two skates together in what direction and with what speed do they move after the collision draw and label a momentum vector diagram show
Well, when two skaters collide, it becomes quite a "skatesy" situation! Let's break it down and have some fun:
First, we have Skater N moving north at 7.8 m/s. Let's call this momentum vector "N".
N (7.8 m/s)
↑
Then, we have Skate E moving east at 3.5 m/s. Let's call this momentum vector "E".
E (3.5 m/s) →
Now, when they collide, it's like a dancefloor collision! They combine their momentums and tango towards a new direction. Let's find out what that is:
To calculate the resultant momentum, we need to add the two vectors mathematically:
Resultant momentum vector = N + E
Since N is moving north and E is moving east, we can imagine it as a right-angled triangle. Now, time for some math magic:
Pythagoras would be happy! We'll use the Pythagorean theorem to find the magnitude of the resultant momentum:
Magnitude = √(N^2 + E^2)
Magnitude = √((7.8 m/s)^2 + (3.5 m/s)^2)
Magnitude ≈ √(60.84 m^2/s^2 + 12.25 m^2/s^2)
Magnitude ≈ √(73.09 m^2/s^2)
Magnitude ≈ 8.55 m/s
So, the magnitude of the resulting momentum is approximately 8.55 m/s. Now, let's find the direction:
We can use some trigonometry to determine the angle between the resultant momentum and the horizontal direction. (Let's say the angle is θ)
θ = tan^(-1)(E/N)
θ = tan^(-1)(3.5 m/s / 7.8 m/s)
θ ≈ 25.80 degrees
So, after the collision, the skaters move in a direction approximately 25.80 degrees north of east, with a speed of 8.55 m/s. Voila! Our momentum vector diagram is complete!
I hope this explanation tickled your funny bone while explaining the physics.
To solve this problem, we will use the principles of conservation of momentum.
Step 1: Calculate the momentum of each skater before the collision.
Momentum (p) is given by the equation p = mass (m) × velocity (v).
For skater N:
Mass (m1) = 58.9 kg
Velocity (v1) = 7.8 m/s
Momentum of skater N: p1 = m1 × v1 = 58.9 kg × 7.8 m/s
For skate E:
Mass (m2) = 72.6 kg
Velocity (v2) = 3.5 m/s
Momentum of skate E: p2 = m2 × v2 = 72.6 kg × 3.5 m/s
Step 2: Calculate the total momentum before the collision.
Total momentum before the collision (p1 + p2) = (m1 × v1) + (m2 × v2)
Step 3: Determine the direction and speed of the combined skaters after the collision.
The total momentum before the collision must be equal to the total momentum after the collision, according to the principle of conservation of momentum.
Let the combined speed after the collision be v.
To find the direction and speed of the combined skaters after the collision, we can analyze the components of momentum in the north and east directions.
In the north direction, the net momentum before and after the collision should be zero. This is because the skaters were only moving in the north and east directions before the collision, and after the collision, their motion should be in a different direction.
In the east direction, the net momentum before and after the collision should also be zero.
Now, let's solve for the combined speed and direction after the collision:
m1 × v1 + m2 × v2 = total mass × combined speed
(58.9 kg × 7.8 m/s) + (72.6 kg × 3.5 m/s) = (58.9 kg + 72.6 kg) × v
Solving this equation will give us the combined speed (v).
Step 4: Draw the momentum vector diagram.
To draw the momentum vector diagram:
- Draw a horizontal line to represent the east direction.
- Draw a vertical line to represent the north direction.
- Label the momentum vectors of skater N and skate E before the collision according to their magnitudes and directions.
- Add the momentum vector of the combined skaters after the collision, labeled with its magnitude and direction.
Note: Without the exact values of mass and velocities, the actual calculations cannot be performed, but the steps provided here constitute the method to solve the problem.
To determine the direction and speed in which the skaters move after the collision, we need to analyze the principles of conservation of momentum.
1. Calculate the initial momentum for each skater:
Initial momentum of skater N (in the north direction) = mass × velocity = 58.9 kg × 7.8 m/s = 458.82 kg·m/s (north)
Initial momentum of skate E (in the east direction) = mass × velocity = 72.6 kg × 3.5 m/s = 253.5 kg·m/s (east)
2. Since momentum is conserved in all directions in a collision, the total initial momentum is equal to the total final momentum. Therefore:
Total initial momentum = Total final momentum
3. Break down the final momentum into its north and east components:
Total final momentum (north) + Total final momentum (east)
4. Use the Pythagorean theorem to find the magnitude of the total final momentum:
Magnitude of the total final momentum = √((Total final momentum (north))^2 + (Total final momentum (east))^2)
5. Use trigonometry to find the direction of the total final momentum:
Direction (θ) of the total final momentum = arctan(Total final momentum (north) / Total final momentum (east))
6. Substitute the values from the calculation above to obtain the magnitude and direction of the total final momentum after the collision.
7. Draw a vector diagram by representing the initial momentum vectors of skater N and skate E and labeling them accordingly. Then, draw the total final momentum vector with its magnitude and direction.
Keep in mind that if the collision is perfectly elastic, both momentum and kinetic energy will be conserved. If it is not perfectly elastic, then only momentum is conserved.
Performing the calculations, drawing the vector diagram, and labeling it will provide a visual representation of the movement of the skaters after the collision.
The momentum vectors are
(0,58.9*7.8) and (72.6*3.5,0)
Their sum is
(254.1,459.42)
So, since the combined mass is 58.9+72.6=131.5
the resultant velocity vector is v=p/m
(1.93,3.49)
or 3.99 at E61°N