A frustrated hockey player, at rest on a frictionless ice rink, throws his stick into the crowd. Since you don’t normally go to hockey games, you decide to turn this into a physics problem and calculate the mass of this player. You observe that the stick is thrown with a speed 9.9 m/s at an angle of 29 degrees above the horizontal. With some quick research on your smartphone, you know that a typical professional hockey stick weighs 2.7 kg. If the hockey player recoiled with a speed of 0.6 m/s, what is their mass in kg?
The mass of the hockey player can be calculated using the equation m = (2.7 kg * 9.9 m/s^2) / (0.6 m/s * sin(29 degrees)). This gives a mass of approximately 48.3 kg.
To solve this problem, we can use the principle of conservation of momentum. In a collision, the total momentum before and after the collision remains constant.
Let's define the following variables:
- Mass of the hockey player: m (unknown)
- Mass of the stick: Ms = 2.7 kg (given)
- Speed with which the hockey player recoiled: Vr = 0.6 m/s (given)
- Speed of the stick before being thrown: Vs = 9.9 m/s (given)
- Angle at which the stick is thrown: θ = 29 degrees (given)
We need to calculate the mass of the hockey player (m).
Firstly, let's calculate the momentum of the stick before being thrown. Momentum (p) is defined as the product of an object's mass and its velocity.
The horizontal component of the stick's momentum (ph) is given by:
ph = Ms * Vs * cos(θ)
The vertical component of the stick's momentum (pv) is given by:
pv = Ms * Vs * sin(θ)
Now, let's calculate the momentum of the hockey player after the throw. Since the hockey player recoiled, the direction of the player's momentum will be opposite to the direction of the stick's momentum:
The horizontal component of the player's momentum (ph') is given by:
ph' = m * Vr
Since there is no vertical movement mentioned in the problem, the vertical component of the player's momentum will be zero.
According to the conservation of momentum, the total momentum before and after the throw should be equal. Therefore, the horizontal components of both the stick and the player's momentum should be equal:
ph = ph'
Ms * Vs * cos(θ) = m * Vr
Now we can solve for the mass of the hockey player (m):
m = (Ms * Vs * cos(θ)) / Vr
Plugging in the given values:
m = (2.7 kg * 9.9 m/s * cos(29°)) / 0.6 m/s
Using a calculator, we find:
m ≈ 42.3 kg
Therefore, the mass of the hockey player is approximately 42.3 kg.
To solve this problem, we can use the principle of conservation of momentum. The total momentum before the throw should be equal to the total momentum after the throw.
1. First, let's find the momentum of the stick before the throw. The momentum of an object is given by the formula:
Momentum = mass × velocity
Given that the stick weighs 2.7 kg and is thrown with a speed of 9.9 m/s, we can calculate its momentum:
Momentum of stick = 2.7 kg × 9.9 m/s = 26.73 kg·m/s
2. Next, let's find the horizontal and vertical components of the stick's velocity. We can use trigonometry to find these components:
Horizontal velocity = velocity × cos(angle)
Vertical velocity = velocity × sin(angle)
Given that the angle of the throw is 29 degrees, we can calculate the horizontal and vertical velocities:
Horizontal velocity = 9.9 m/s × cos(29 degrees) = 8.708 m/s
Vertical velocity = 9.9 m/s × sin(29 degrees) = 4.811 m/s
3. The total horizontal momentum before the throw should be equal to the total horizontal momentum after the throw. Since there is no external force acting on the hockey player-stick system in the horizontal direction (due to a frictionless surface), we can set up the equation:
(Mass of player + Mass of stick) × Recoil velocity = Mass of stick × Horizontal velocity
Given that the recoil velocity is 0.6 m/s, we can rewrite the equation as:
(Mass of player + 2.7 kg) × 0.6 m/s = 2.7 kg × 8.708 m/s
4. Now, let's solve the equation for the mass of the player:
(Mass of player + 2.7 kg) × 0.6 m/s = 2.7 kg × 8.708 m/s
Mass of player + 2.7 kg = (2.7 kg × 8.708 m/s) / 0.6 m/s
Mass of player + 2.7 kg = 39.282 kg
Mass of player = 39.282 kg - 2.7 kg
Mass of player = 36.582 kg
Therefore, the mass of the frustrated hockey player is approximately 36.582 kg.