A 8.30 kg object with a speed of 34.0 m/s strikes a steel plate at an angle of 45.0 degrees with the normal to the plate, and rebounds at the same speed and angle on the other side of the normal. What is the change (in magnitude only) of the linear momentum of the object (in kilogram-meters/second)?

To find the change in magnitude of the linear momentum of the object, we need to calculate the initial and final magnitudes of the linear momentum and subtract them.

The linear momentum of an object is given by the product of its mass and velocity:

Linear Momentum = Mass × Velocity

Given that the object has a mass of 8.30 kg and a speed of 34.0 m/s, we can calculate the initial linear momentum:

Initial Momentum = 8.30 kg × 34.0 m/s = 282.2 kg·m/s

Now, since the object rebounds with the same speed and angle on the other side of the normal, its final momentum will have the same magnitude.

Therefore, the change in magnitude of the linear momentum is:

Change in Momentum = Final Momentum - Initial Momentum
= |Final Momentum| - |Initial Momentum|

Since both the final and initial momenta have the same magnitude,

Change in Momentum = |Final Momentum| - |Initial Momentum|
= |Final Momentum| - |282.2 kg·m/s|

To find the final momentum, we need to decompose the momentum into horizontal and vertical components, using the given angle of 45 degrees. Since the angle is the same on both sides of the normal, both the horizontal and vertical components of velocity remain the same after the collision.

The horizontal component of velocity is given by:

Horizontal Velocity = Velocity × cos(angle)

Horizontal Velocity = 34.0 m/s × cos(45°) = 34.0 m/s × (sqrt(2)/2) = 34.0 m/s × 0.7071 = 24.08 m/s

The vertical component of velocity is given by:

Vertical Velocity = Velocity × sin(angle)

Vertical Velocity = 34.0 m/s × sin(45°) = 34.0 m/s × (sqrt(2)/2) = 34.0 m/s × 0.7071 = 24.08 m/s

As mentioned earlier, the final momentum will have the same magnitude as the initial momentum. Therefore, the final momentum is:

Final Momentum = Mass × Final Velocity

The final velocity can be calculated using the horizontal and vertical velocities:

Final Velocity = sqrt((Horizontal Velocity)^2 + (Vertical Velocity)^2)

Final Velocity = sqrt((24.08 m/s)^2 + (24.08 m/s)^2) = sqrt(579.26 m^2/s^2) = 24.08 m/s

Now we can calculate the final momentum:

Final Momentum = 8.30 kg × 24.08 m/s = 199.82 kg·m/s

Finally, we can find the change in magnitude of the linear momentum:

Change in Momentum = |Final Momentum| - |Initial Momentum|
= |199.82 kg·m/s| - |282.2 kg·m/s|
= 199.82 kg·m/s - 282.2 kg·m/s
= -82.38 kg·m/s (Note: The negative sign indicates a change in direction)

Therefore, the change in magnitude of the linear momentum of the object is 82.38 kg·m/s.