A hockey player is moving at 7.00 m/s when he hits the puck toward the goal. The speed of the puck relative to the player is 32.0 m/s. The line between the center of the goal and the player makes a 90° angle relative to his path as shown in Figure 3.30. What angle must the puck's velocity make relative to the player (in his frame of reference) to hit the center of the goal?
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To determine the angle at which the puck's velocity must be set relative to the player's frame of reference, we can use the concept of relative velocity.
Here's how we can approach the problem step by step:
1. Start by understanding the given information:
- The hockey player's velocity (v_player) = 7.00 m/s
- The velocity of the puck relative to the player (v_puck_relative) = 32.0 m/s
2. Draw a diagram to visualize the situation:
- Draw a line representing the player's path and another line representing the line between the center of the goal and the player.
----[Player's path]
|
|
|
|--------[Line to the center of the goal]
3. Identify the known velocities:
- The puck's velocity relative to the player (v_puck_relative) = 32.0 m/s.
- The player's velocity (v_player) = 7.00 m/s.
4. Use the concept of relative velocity to find the puck's velocity relative to the ground (frame of reference):
- The puck's velocity relative to the ground (v_puck_ground) = v_player + v_puck_relative.
v_puck_ground = v_player + v_puck_relative
= 7.00 m/s + 32.0 m/s
= 39.0 m/s
5. Use trigonometry to find the angle the puck's velocity makes with the line to the center of the goal:
- We are given that the line between the player and the goal makes a 90° angle with the player's path.
- In a right-angled triangle, the tangent of an angle is equal to the ratio of the lengths of the opposite side and the adjacent side.
- In this case, the opposite side is the component of the puck's velocity perpendicular to the player's path, and the adjacent side is the component of the puck's velocity parallel to the player's path.
- Therefore, the angle can be found using the equation:
Tan(angle) = (Opposite side / Adjacent side)
- The opposite side is the component of the puck's velocity perpendicular to the player's path, which is v_puck_relative.
- The adjacent side is the component of the puck's velocity parallel to the player's path, which is v_puck_ground.
- Tan(angle) = (v_puck_relative / v_puck_ground)
6. Substitute the values into the equation and solve for the angle:
- Tan(angle) = (32.0 m/s / 39.0 m/s)
- angle = arctan(32.0 m/s / 39.0 m/s)
Using a calculator or table, we find that the angle is approximately 41.0°.
Therefore, the puck's velocity must be set at an angle of approximately 41.0° relative to the player's frame of reference to hit the center of the goal.