(I) In a ballistic pendulum experiment, projectile 1 results in a maximum height f> of the pendulum equal to 2.6 cm. A second projectile (of the same mass) causes the pendulum to swing twice as high, h2 =5.2 cm. The second projectile was how many times faster than the first?
We have to obtain the relation between the velocities of projectile in both the situation.
Step 1 of 5</p>
The ballistic pendulum is a device used to measure the speed of the projectile. In this device, the projectile of mass is fired into a large block of mass , which is suspended like a pendulum. As a result of collision the projectile get embedded into the pendulum and the pendulum and projectile together swing up to a maximum height . We can analyze the process by using law of conservation of momentum and law of conservation of energy.
Step 2 of 5</p>
Law of conservation of linear momentum: When no external force acts on the system, then the final momentum of the system is equal to their initial momentum.
Law of conservation of Energy: Energy can neither be created nor be destroyed. It can only be converted from one form into other. The total amount of energy in an isolated system remains constant.
Step 3 of 5</p>
We choose , when the pendulum hangs vertically, and then when the pendulum-projectile system reaches its maximum height .
We choose the bottom most of pendulum as zero potential energy.
Thus the speed of the projectile with which it was moving, is given as,
Using equation (1), the horizontal speed of the first projectile which attained the maximum height is
Using equation (1), the horizontal speed of the second projectile which attained the maximum height is