A bullet (m = 0.0250 kg) is fired with a speed of 91.00 m/s and hits a block (M = 2.30 kg) supported by two light strings as shown, stopping quickly. Find the height to which the block rises.
and
Find the angle (in degrees) through which the block rises, if the strings are 0.260 m in length.
You do not describe the figure. Why are two strings needed to hold the block?
In general, use conservation of momentum to calculate the momentum of and velocity V' the block after impact, but before it starts to swing upwards. Calculate also its initial kinetic energy at that time, with the bullet inside. If the buolet velcoity is V,
m V = (M+m) V'
The initial kinetic energy
(1/2)(M+m)V'^2 is equal to the potential enewrgy increase (M+m)gH after is swings upwards a distance H. Use trigonometry to get the angle.
To solve this problem, we can use the principle of conservation of mechanical energy. We can equate the initial kinetic energy of the bullet to the final potential energy of the block.
1. Find the initial kinetic energy (KE) of the bullet:
The formula for kinetic energy is KE = 0.5 * mass * velocity^2.
Substituting the given values, we get:
KE = 0.5 * 0.0250 kg * (91.00 m/s)^2. Calculate this to find the initial kinetic energy.
2. Find the final potential energy (PE) of the block:
The formula for potential energy is PE = mass * gravity * height.
Substituting the given values, we have:
PE = 2.30 kg * 9.8 m/s^2 * height. Rearrange this equation to solve for height.
3. Equate the initial kinetic energy to the final potential energy:
Since energy is conserved, we have KE = PE.
Set the equations for KE and PE equal to each other and solve for height.
Once you find the height to which the block rises, you can then proceed to find the angle through which the block rises.
4. Find the angle (θ) through which the block rises:
The angle can be found using trigonometry. The length of the strings is given as 0.260 m, which represents the hypotenuse of a right triangle formed by the horizontal displacement and the vertical displacement.
The sine of the angle is given by the ratio of the vertical displacement to the hypotenuse: sin(θ) = height / 0.260 m.
5. Solve for the angle:
Use inverse sine (sin^-1) to find the angle θ. Calculate the inverse sine of the ratio found in the previous step to obtain the angle in degrees.
By following these steps, you can solve the problem and find both the height to which the block rises and the angle through which it rises.