The figure shows a 0.3 kg baseball just before and just after it collides with a bat. Just before, the ball has velocity of magnitude 11.4 m/s and angle = 31.4 degrees. Just after, it is traveling directly upward with velocity of magnitude 9.00 m/s. The duration of the collision is 1.90 ms. What are the (a) magnitude and (b) direction (relative to the positive direction of the x axis) of the impulse on the ball from the bat?
To determine the magnitude and direction of the impulse on the ball from the bat, we need to calculate the change in momentum.
The momentum of an object is given by the equation:
p = m * v
Where:
p = momentum
m = mass
v = velocity
In this case, the momentum of the ball just before the collision is:
p1 = m * v1
And the momentum of the ball just after the collision is:
p2 = m * v2
The impulse on the ball, I, can be calculated by subtracting the initial momentum from the final momentum:
I = Δp = p2 - p1
To calculate the magnitude of the impulse, we take the absolute value of the impulse:
|I| = |p2 - p1|
To find the direction of the impulse, we need to calculate the angle it makes with the positive x-axis. We can calculate this angle using trigonometry.
Let's break down the steps to calculate the magnitude and direction of the impulse:
Step 1: Convert the angle from degrees to radians:
θ_rad = θ_deg * (π / 180)
Step 2: Calculate the x and y components of the initial velocity:
v1x = v1 * cos(θ_rad)
v1y = v1 * sin(θ_rad)
Step 3: Calculate the y component of the final velocity:
v2y = v2
Step 4: Calculate the change in momentum in the x direction:
Δpx = m * (0 - v1x)
Step 5: Calculate the change in momentum in the y direction:
Δpy = m * (v2y - v1y)
Step 6: Calculate the magnitude of the impulse:
|I| = sqrt(Δpx^2 + Δpy^2)
Step 7: Calculate the direction of the impulse:
θ_I = atan2(Δpy, Δpx)
Now, let's plug in the given values and calculate the magnitude and direction of the impulse.