A hockey player lobs a puck at an empty net with an initial velocity of 15.0 m/s at an angle of 23.0°. What is the maximum height the puck reaches?
7.97 m
1.75 m
3.22 m
6.51 m
To find the maximum height the puck reaches, we first need to break down the initial velocity into its horizontal and vertical components. The horizontal component will remain constant throughout the motion, while the vertical component will be affected by gravity.
The horizontal component (Vx) can be calculated using the initial velocity and the given angle:
Vx = V * cos(angle)
= 15.0 m/s * cos(23.0°)
≈ 13.82 m/s
The vertical component (Vy) can be calculated in a similar way:
Vy = V * sin(angle)
= 15.0 m/s * sin(23.0°)
≈ 6.12 m/s
At the highest point of the projectile's trajectory, the vertical component of velocity will be zero. This is because the puck will momentarily stop moving vertically before starting to descend due to the pull of gravity.
To find the time it takes for the puck to reach its maximum height, we can use the equation of motion:
Vy = Vy0 + (-g) * t
Where:
- Vy is the vertical component of velocity at any given time (in this case, at the highest point)
- Vy0 is the initial vertical component of velocity (6.12 m/s)
- g is the acceleration due to gravity (9.8 m/s², assuming Earth's gravity)
- t is the time it takes to reach the maximum height
Since Vy = 0 at the highest point, we can rearrange the equation to solve for t:
0 = 6.12 m/s - 9.8 m/s² * t
Solving for t:
t = 6.12 m/s / 9.8 m/s²
≈ 0.625 s
Now that we have the time it takes to reach the maximum height, we can use it to find the maximum height (h). The formula for calculating the height is:
h = Vy0 * t + (1/2) * (-g) * t²
Plugging in the values:
h = 6.12 m/s * 0.625 s + (1/2) * (-9.8 m/s²) * (0.625 s)²
≈ 1.76 m
Therefore, the maximum height the puck reaches is approximately 1.76 m. Based on the given answer choices, the closest option is 1.75 m.