A 5 kg object moving with a speed of 6.6 m/s collides with an 18 kg object moving with a velocity of 8.4 m/s in a direction 19 degrees from the initial direction of the motion of the 5 kg object. What is the speed of the two objects after collision if they remain stuck together? Answer in m/s What is the change in direction experienced by the lighter of the two object? Answer in units of degrees.
Find initial momentum for the smaller object, via p1 = mv. Think of this momentum as being in the x-direction. Now break up the other objects momentum into its x and y components, p2x = mvcos(theta) and p2y = mvsin(theta). Final momentum in x-direction = p1+p2x and final momentum in y-direction = p2y. Use these components to find the momentum in the actual direction of the motion. Divide this momentum by the combined mass of the objects in order to find v. In order to find the change in direction, take the inverse tan of (p2y)/(p1+p2x). This is for the AP Physics students of decades to come.
Well, well, well. Looks like we've got a physics problem on our hands! Let's dive in, shall we?
First, we need to calculate the total momentum before the collision. That's just the sum of the momenta of the two objects.
The momentum of an object is given by the product of its mass and velocity. For the 5 kg object, its momentum is 5 kg * 6.6 m/s = 33 kg·m/s. Meanwhile, the 18 kg object has a momentum of 18 kg * 8.4 m/s = 151.2 kg·m/s.
To find the total momentum, we add these two values together: 33 kg·m/s + 151.2 kg·m/s = 184.2 kg·m/s.
Now, since the objects stick together after the collision, their masses combine to form a single object. The new mass is the sum of the masses: 5 kg + 18 kg = 23 kg.
To find the final velocity, we divide the total momentum by the new mass: 184.2 kg·m/s ÷ 23 kg = 8 m/s (approximately).
Therefore, the speed of the two objects, once stuck together, is 8 m/s (approximately).
Now, let's move on to the change in direction experienced by the lighter object.
We are given that the 5 kg object initially moved in a direction 19 degrees away from the velocity of the 18 kg object. After the collision, the lighter object will move in the same direction as the heavier object. So, the lighter object underwent a change in direction of 19 degrees!
Ta-da! Now you have the answers. Hope I managed to clown around with physics a little bit to brighten your day!
To find the final velocity of the objects after collision, we can use the principle of conservation of momentum.
Step 1: Calculate the initial momentum of each object:
- Momentum = mass x velocity
For the 5 kg object:
Initial momentum = mass x velocity
= 5 kg x 6.6 m/s
= 33 kg·m/s
For the 18 kg object:
Initial momentum = mass x velocity
= 18 kg x 8.4 m/s
= 151.2 kg·m/s
Step 2: Calculate the horizontal and vertical components of momentum for the 5 kg object:
- Horizontal component = momentum x cos(angle)
- Vertical component = momentum x sin(angle)
For the 5 kg object:
Horizontal component = 33 kg·m/s x cos(0 degrees)
= 33 kg·m/s x 1
= 33 kg·m/s
Vertical component = 33 kg·m/s x sin(0 degrees)
= 33 kg·m/s x 0
= 0 kg·m/s
Step 3: Calculate the horizontal and vertical components of momentum for the 18 kg object:
- Horizontal component = momentum x cos(angle)
- Vertical component = momentum x sin(angle)
For the 18 kg object:
Horizontal component = 151.2 kg·m/s x cos(19 degrees)
≈ 145.9717 kg·m/s
Vertical component = 151.2 kg·m/s x sin(19 degrees)
≈ 50.5957 kg·m/s
Step 4: Calculate the total horizontal and vertical momentum after the collision:
- Total horizontal momentum = sum of the horizontal components
- Total vertical momentum = sum of the vertical components
Total horizontal momentum = 33 kg·m/s + 145.9717 kg·m/s
≈ 178.9717 kg·m/s
Total vertical momentum = 0 kg·m/s + 50.5957 kg·m/s
≈ 50.5957 kg·m/s
Step 5: Calculate the final momentum after the collision:
- Final momentum = total horizontal momentum + total vertical momentum
Final momentum = √(total horizontal momentum^2 + total vertical momentum^2)
= √((178.9717 kg·m/s)^2 + (50.5957 kg·m/s)^2)
≈ √(32052.3201 kg^2·m^2/s^2 + 2560.8436 kg^2·m^2/s^2)
≈ √(34613.1637 kg^2·m^2/s^2)
≈ 185.99 kg·m/s
Step 6: Calculate the final velocity of the objects:
- Final velocity = final momentum / total mass
Total mass = mass of the 5 kg object + mass of the 18 kg object
= 5 kg + 18 kg
= 23 kg
Final velocity = 185.99 kg·m/s / 23 kg
≈ 8.087 m/s
Therefore, the speed of the two objects after collision, when they remain stuck together, is approximately 8.087 m/s.
To find the change in direction experienced by the lighter object, we can use trigonometry.
Step 1: Calculate the change in direction using the inverse tangent function:
- Change in direction = arctan(vertical component / horizontal component)
Change in direction = arctan(50.5957 kg·m/s / 33 kg·m/s)
≈ arctan(1.5359)
≈ 57.38 degrees
Therefore, the change in direction experienced by the lighter of the two objects is approximately 57.38 degrees.
To find the speed of the two objects after they collide, we need to apply the principle 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: momentum = mass * velocity.
Let's calculate the initial momentum of the 5 kg object (object A) and the 18 kg object (object B):
Momentum of A before collision = mass of A * velocity of A
= 5 kg * 6.6 m/s
= 33 kg m/s
Momentum of B before collision = mass of B * velocity of B
= 18 kg * 8.4 m/s
= 151.2 kg m/s
Now, since the two objects stick together after the collision, their combined mass will be the sum of their individual masses:
Total mass after collision = mass of A + mass of B
= 5 kg + 18 kg
= 23 kg
Since the two objects stick together, their combined momentum after the collision will be equal to the total momentum before the collision. Therefore:
Total momentum after collision = Total momentum before collision
= 33 kg m/s + 151.2 kg m/s
= 184.2 kg m/s
To find the speed of the two objects after the collision, we divide the combined momentum by the combined mass:
Speed after collision = Total momentum after collision / Total mass after collision
= 184.2 kg m/s / 23 kg
≈ 8.01 m/s
Therefore, the speed of the two objects after the collision, when they remain stuck together, is approximately 8.01 m/s.
Now, let's move on to calculating the change in direction experienced by the lighter of the two objects (object A). Since the two objects stick together, their combined velocity after the collision will be the same for both objects. We can use trigonometry to find the angle between their initial direction and their final direction.
Let's consider the angle between the initial direction of object A and the final direction of the combined objects. According to the problem statement, this angle is 19 degrees.
To find the change in direction experienced by object A, we subtract this angle from 180 degrees:
Change in direction = 180 degrees - 19 degrees
= 161 degrees
Therefore, the change in direction experienced by the lighter object (object A) is 161 degrees.