An object of mass 540 gram has an initial velocity of -0.4i+0.6j m/s. It collides with and sticks to another object of mass 880 gram moving with an initial velocity of 7.8i+2.3j+1.3k m/s (here i, j, and k are unit vectors along the x-, y-, and z-axis respectively). Find the speed of the composite object.
To find the speed of the composite object, we need to calculate the final velocity of the composite object after the collision.
First, we need to determine the initial momentum of each object. Momentum is defined as the product of mass and velocity.
For the first object:
Mass (m1) = 540 grams = 0.54 kg
Initial velocity (v1) = -0.4i + 0.6j m/s
The momentum of the first object is given by:
Momentum (p1) = m1 * v1
Next, let's calculate the initial momentum of the second object:
Mass (m2) = 880 grams = 0.88 kg
Initial velocity (v2) = 7.8i + 2.3j + 1.3k m/s
The momentum of the second object is given by:
Momentum (p2) = m2 * v2
Since the objects stick together after the collision, we can add their momenta to obtain the total initial momentum (ptotal) of the composite object:
ptotal = p1 + p2
Now, we can calculate the mass (m) and velocity (v) of the composite object using the total initial momentum.
m = m1 + m2
v = ptotal / m
Finally, to find the speed of the composite object, we need to calculate the magnitude of the velocity vector.
Speed = |v|
To summarize, the steps to find the speed of the composite object after the collision are as follows:
1. Calculate the initial momentum (p1) of the first object.
2. Calculate the initial momentum (p2) of the second object.
3. Calculate the total initial momentum (ptotal) of the composite object.
4. Calculate the mass (m) and velocity (v) of the composite object using ptotal.
5. Calculate the speed of the composite object as the magnitude of the velocity vector.
I will now perform the calculations for you.