An outfielder throws a 1.88 kg baseball at a
speed of 57 m/s and an initial angle of 10.2�.
What is the kinetic energy of the ball at the
highest point of its motion?
Answer in units of J
To find the KE of the ball at the highest ball of motion, we have to use the initial velocity and angle to find the horizontal velocity.
V(horizontal) = V(initial) * cos(theta)
V(horizontal) = (57 m/s) * cos(10.2)
V(horizontal) = 56.099 m/s
Using the velocity in the horizontal direction that we found, along with the mass given, we can calculate the KE of the ball at its highest point.
KE = (1/2)mv^2
KE = (1/2)(1.88 kg)(56.099 m/s)^2
KE = 2958.27 J
To find the kinetic energy of the baseball at the highest point of its motion, we need to first determine the highest point of the baseball's trajectory. At the highest point, the vertical component of the velocity will be zero, meaning that the baseball will be momentarily at rest before starting to fall back down.
To determine the highest point, we can use kinematic equations. We know that the initial vertical velocity is given by:
Vy(initial) = V(initial) * sin(θ)
where V(initial) is the initial velocity (57 m/s) and θ is the initial angle (10.2 degrees).
Next, we can use the kinematic equation for vertical motion to find the time it takes for the baseball to reach the highest point. The equation is:
Vy(final) = Vy(initial) - g * t
Since the baseball is at rest at the highest point, Vy(final) is 0. The acceleration due to gravity (g) is approximately 9.8 m/s^2. Solving for t, we get:
0 = V(initial) * sin(θ) - g * t
Rearranging the equation to solve for t, we have:
t = V(initial) * sin(θ) / g
Now we can find the time it takes for the baseball to reach the highest point by plugging in the given values:
t = (57 m/s) * sin(10.2 degrees) / 9.8 m/s^2
Calculating this, we find t ≈ 0.96 seconds.
Next, we can find the height of the highest point using the equation:
y = Vy(initial) * t - 0.5 * g * t^2
Since the initial vertical velocity is given by Vy(initial) = V(initial) * sin(θ), we have:
y = V(initial) * sin(θ) * t - 0.5 * g * t^2
Substituting the known values:
y = (57 m/s) * sin(10.2 degrees) * 0.96 s - 0.5 * (9.8 m/s^2) * (0.96 s)^2
Performing the calculations, we find y ≈ 2.56 meters.
At this highest point, the kinetic energy of the baseball is zero, since it is momentarily at rest. Therefore, the kinetic energy at the highest point of its motion is 0 J (zero joules).