1.
A football player weighing 75 kg running at 2 m/s toward west tackles a 70 kg player running at 1.5 m/s in the opposite direction. What is the final velocity of the players if they both fall together?
5.5
hard
To find the final velocity of the players when they fall together, we need to use the concept of conservation of momentum. Momentum is a vector quantity, which means it has both magnitude and direction.
The formula for momentum is given by:
Momentum = mass × velocity
Let's assume that the positive direction is towards the west, where the first player is running. The second player is running towards the east.
Step 1: Calculate the initial momentum of each player.
Player 1 (west):
Initial momentum = mass × velocity
Momentum of player 1 = 75 kg × 2 m/s (towards west)
Momentum of player 1 = 150 kg·m/s (towards west)
Player 2 (east):
Initial momentum = mass × velocity
Momentum of player 2 = 70 kg × (-1.5 m/s) (towards east; we use negative sign to indicate opposite direction)
Momentum of player 2 = -105 kg·m/s (towards east)
Step 2: Calculate the total initial momentum of the system.
Total initial momentum = Momentum of player 1 + Momentum of player 2
Total initial momentum = 150 kg·m/s (towards west) + (-105 kg·m/s) (towards east)
Total initial momentum = 45 kg·m/s towards the west.
Step 3: Calculate the final velocity of the players when they fall together.
Since the players fall together, their final momentum will be zero (as their total momentum after the collision becomes zero).
Total final momentum = 0 kg·m/s
Using the formula for momentum:
Total final momentum = Momentum of player 1 + Momentum of player 2
0 kg·m/s = Mass of player 1 × Velocity of player 1 + Mass of player 2 × Velocity of player 2
To find the final velocity when the total final momentum is zero, we have:
0 = 75 kg × final velocity (vf) + 70 kg × final velocity (vf)
Simplifying the equation:
0 = 145 kg × vf
Divide both sides by 145 kg:
vf = 0 m/s
Therefore, the final velocity of the players when they fall together is 0 m/s, which means they come to a complete stop.