Skater N, 58.9 kg, is moving north at a speed of 7.8 m/s when she collides with Skater E, 72.6 kg, moving east at 3.5 m/s. The two skaters are stuck together. In what direction and with what speed do they move after the collision?
Well, this is quite the skating adventure! Let me see if I can crunch some numbers for you.
Skater N, weighing in at 58.9 kg, is moving north at a velocity of 7.8 m/s. Skater E, on the other hand, weighs 72.6 kg and is skating east at 3.5 m/s.
When they collide, they merge into a super-skater duo. To find out which direction they'll be skating, we'll need to apply some physics magic.
First, let's break their velocities down into components. Skater N's velocity is purely vertical, while Skater E's velocity is purely horizontal.
When they collide, they become stuck together, forming a combined mass of 58.9 kg + 72.6 kg = 131.5 kg.
To find their final speed, we can use the principle of conservation of momentum, which states that the total momentum before the collision should equal the total momentum after the collision.
The total momentum before the collision can be calculated by multiplying the individual masses by their respective velocities. For Skater N, momentum = 58.9 kg * 7.8 m/s = 459.42 kg·m/s. For Skater E, momentum = 72.6 kg * 3.5 m/s = 253.5 kg·m/s.
After the collision, this total momentum will remain the same. Since the two skaters are now stuck together, their combined momentum will be constant.
We can find their final speed using the formula:
combined momentum = (combined mass) * (final speed)
So, 459.42 kg·m/s + 253.5 kg·m/s = 131.5 kg * final speed.
Solving for the final speed, we get: final speed = (459.42 kg·m/s + 253.5 kg·m/s)/131.5 kg ≈ 5.99 m/s.
To determine the direction of their movement, we can use basic trigonometry. Since Skater N was moving purely north and Skater E was moving purely east, the resulting direction of their movement can be found using tangent:
tan(theta) = (vertical velocity) / (horizontal velocity).
tan(theta) = 7.8 m/s / 3.5 m/s ≈ 2.23.
Solving for theta, we find that theta ≈ 66.87 degrees.
So, after the collision, the super-skater duo will be moving at approximately 5.99 m/s in a direction approximately 66.87 degrees northeast. It seems like they'll have a new skating routine to impress the crowd!
To determine the direction and speed at which the two skaters move after the collision, we can use the principles of conservation of momentum. The total momentum before the collision should be equal to the total momentum after the collision.
The momentum of an object is defined as the product of its mass and velocity. Mathematically, momentum (p) is given by the equation:
p = m * v
For Skater N before the collision:
pN = mN * vN
= 58.9 kg * 7.8 m/s
For Skater E before the collision:
pE = mE * vE
= 72.6 kg * 3.5 m/s
Since the skaters are stuck together after the collision, their combined mass is the sum of their individual masses:
m = mN + mE
= 58.9 kg + 72.6 kg
To find the direction, we can use vector addition by considering the velocities as vectors. Since Skater N is moving north and Skater E is moving east, we can consider their velocities as components in the x and y directions using a coordinate system.
The x-component of the velocity after the collision will remain the same, as there are no external forces acting in the horizontal direction.
For the x-component:
v_x = vE_x
= 3.5 m/s
The y-component of the velocity after the collision will be the sum of the y-components of the velocities before the collision.
For the y-component:
v_y = vN_y + vE_y
= vN + 0 (since Skater N is moving only in the y-direction)
Now, we can find the total linear momentum of the system after the collision (p'):
p' = m * v'
= (58.9 kg + 72.6 kg) * sqrt(v_x^2 + v_y^2)
Since momentum is conserved, the total momentum before the collision (pN + pE) should be equal to the total momentum after the collision (p'):
pN + pE = p'
Now, we can rearrange the equation to solve for the magnitude of v':
v' = sqrt((pN + pE)^2 / m^2 - v_x^2)
To find the direction of the velocity vector after the collision, we can use trigonometry. The angle θ can be calculated as:
θ = arctan(v_y / v_x)
Substituting the given values and solving the equations will provide us with the final answer, which is the direction and magnitude of the velocity at which the skaters move after the collision.