An outfielder throws a 0.150-kg baseball at a speed of 50.0 m/s and an initial angle of 30.0°. What is the kinetic energy of the ball at the highest point of its motion?
horizontal velocity stays the same.
KE= 1/2 m (Vh)^2 where Vh= 50cos50
To find the kinetic energy of the ball at the highest point of its motion, we need to use the concept of projectile motion. In projectile motion, the vertical component of motion is independent of the horizontal component.
Given:
Mass of the baseball, m = 0.150 kg
Initial velocity, v₀ = 50.0 m/s
Initial angle, θ = 30.0°
To find the maximum height, we can first find the vertical component of the initial velocity using the sine function:
v₀y = v₀ * sin(θ)
v₀y = 50.0 m/s * sin(30.0°)
v₀y ≈ 25.0 m/s
Next, we can find the maximum height using the following equation of motion:
v² = u² + 2as
where
v = final velocity (at highest point) = 0 m/s (since the ball momentarily stops at the highest point)
u = initial velocity (vertical component) = v₀y = 25.0 m/s
a = acceleration = -9.8 m/s² (assuming the negative direction is upwards)
s = displacement (maximum height)
0 = (25.0 m/s)² + 2 (-9.8 m/s²) s
0 = 625 m²/s² - 19.6 m/s² s
19.6 m/s² s = 625 m²/s²
s = 625 m²/s² / 19.6 m/s²
s ≈ 31.89 m
Now, we can calculate the kinetic energy of the ball at the highest point, using the formula:
kinetic energy = 0.5 * m * v²
where
m = mass of the baseball = 0.150 kg
v = velocity (at highest point) = v₀y = 25.0 m/s
kinetic energy = 0.5 * (0.150 kg) * (25.0 m/s)²
kinetic energy ≈ 93.75 J
Therefore, the kinetic energy of the baseball at the highest point of its motion is approximately 93.75 Joules.
To calculate the kinetic energy of the baseball at the highest point of its motion, we need to first determine the velocity of the baseball at that point.
1. Break down the initial velocity vector: The initial velocity of the baseball can be broken down into its horizontal and vertical components.
- The horizontal component (Vx) can be calculated as V * cos(θ), where V is the magnitude of the initial velocity (50.0 m/s) and θ is the initial angle (30.0°).
- The vertical component (Vy) can be calculated as V * sin(θ), where V is the magnitude of the initial velocity (50.0 m/s) and θ is the initial angle (30.0°).
2. Determine the time taken to reach the highest point: At the highest point of its motion, the vertical component of velocity becomes zero. Use this fact to calculate the time (t) taken for the baseball to reach the highest point. Since gravity acts vertically downwards, the time taken to reach the highest point will be the same as the time taken for the vertical component of velocity to decrease to zero.
- The equation to determine the time taken for an object to reach its highest point is t = Vy / g, where g is the acceleration due to gravity (9.8 m/s²).
3. Calculate the horizontal velocity at the highest point: The horizontal component of velocity remains constant throughout the motion. Therefore, the horizontal velocity at the highest point is the same as the initial horizontal component (Vx).
4. Determine the kinetic energy at the highest point: The kinetic energy (KE) of an object can be calculated using the equation KE = (1/2) * m * v², where m is the mass of the object and v is its velocity.
- Since the baseball is at the highest point of its motion, its vertical component of velocity is zero. Therefore, only the horizontal component contributes to the velocity. The velocity at the highest point is equal to the horizontal component of velocity (Vx).
- Substitute the mass of the baseball (0.150 kg) and the horizontal component of velocity (Vx) into the kinetic energy equation to calculate the result.
By following these steps, you can calculate the kinetic energy of the ball at the highest point of its motion.