three dimensional identical balls 1,2 and 3 are placed on a straight line at a separation of 10m between balls . initially they are at rest. ball 1 is given a velocity of 10m/s towards ball 2.
collision between ball 1 and 2 is inelastic with e=0.5 . but collision between ball 2 and 3 is elastic. what is the time interval between 2 consecutive colisions between ball 1 and 2?
To find the time interval between two consecutive collisions between ball 1 and ball 2, we need to calculate the time it takes for ball 1 to reach ball 2 after the first collision.
Let's break down the problem into two parts:
1. Collision between Ball 1 and Ball 2:
Since the collision is inelastic with a coefficient of restitution (e) of 0.5, we know that the relative velocity after the collision is half of the relative velocity before the collision.
Given:
Mass of Ball 1 = Mass of Ball 2 (since they are identical)
Initial velocity of Ball 1 (u1) = 10 m/s
To find the velocity of Ball 2 after the collision (v2), we can use the formula:
v2 = ((e + 1) * u1) / 2
In our case, e = 0.5:
v2 = ((0.5 + 1) * 10) / 2
v2 = 15 m/s
2. Collision between Ball 2 and Ball 3:
The collision between Ball 2 and Ball 3 is described as elastic, meaning that both momentum and kinetic energy are conserved.
Since mass and initial velocities are the same for both Ball 2 and Ball 3, their velocities after the collision (v2' and v3') will simply be an exchange of velocities:
v2' = v3
v3' = v2
Therefore, after this collision, we have Ball 2 traveling with a velocity of v3, and Ball 3 traveling with a velocity of v2.
Now, the time it takes for Ball 1 to reach Ball 2 after the first collision is equal to the distance (separation) between Ball 1 and Ball 2 divided by the relative velocity:
Time interval = 10m / (v2 - u1)
Time interval = 10m / (15m/s - 10m/s)
Time interval = 10m / 5m/s
Time interval = 2 seconds
So, the time interval between two consecutive collisions between Ball 1 and Ball 2 is 2 seconds.