Lance threw a ball straight up into the air. The ball's gravitational potential energy at its peak height was 0.4 joules, and its speed when it hit the ground was 4 meters per second. What was the mass of the ball?
If the equation you're using is (1/2)M*V^2= J and you know J is o.4 just fill in the velocity(speed) and work it out. Here's how I worked it:
(1/2)M*4^2 = 0.4 (change the numbers around)
(Now it looks like:)
0.4/4^2/(1/2) = M (divide 0.4 by 4^2)
0.4/4^2 = 0.025 (then the answer by 1/2)
0.025/(1/2) = 0.050 or 0.05
so your answer is that the mass of the ball Lance threw is 0.05-kg.
To find the mass of the ball, we can start by using the relationship between gravitational potential energy and the height of the object.
The gravitational potential energy (PE) of an object is given by the formula:
PE = m * g * h
Where m is the mass of the ball, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height of the ball at its peak.
Given that the gravitational potential energy (PE) is 0.4 joules, we can write:
0.4 J = m * 9.8 m/s² * h
At the peak height, the object momentarily stops before falling back down. Therefore, at this point, the ball's velocity is 0 m/s. Using the equation for final velocity (v) in free fall:
v² = u² + 2 * a * h
Where u is the initial velocity (which was thrown straight up) and a is the acceleration due to gravity, we can solve for h:
0 = u² + 2 * 9.8 m/s² * h
Since the ball is thrown straight up, the initial velocity (u) is positive. The final velocity (v) when it hits the ground is given as 4 m/s.
Using the equation for final velocity:
v = u + a * t
Where t is the time it takes for the ball to reach the ground. At the peak height, the velocity is zero, so we can solve for the time:
0 = u + 9.8 m/s² * t
Since the ball is thrown straight up, we can assume the positive direction for upward motion, making the initial velocity (u) positive. The time (t) it takes for the ball to reach the ground is the same as the time it takes for it to reach its peak height, but with opposite direction and acceleration:
t = -u / a
Now we can substitute the obtained time (t) into the equation for height:
0 = u² + 2 * 9.8 m/s² * h
0 = (u² + 2 * 9.8 m/s² * h) * (-u / 9.8 m/s²)
0 = u * (-u / 9.8 m/s²) + 2 * h
0 = -u² / 9.8 m/s² + 2h
Rearranging this equation, we can solve for h:
u² / 9.8 m/s² = 2h
h = u² / (2 * 9.8 m/s²)
Given that the final velocity (v) is 4 m/s and the initial velocity (u) is positive, we can find the initial velocity using the equation:
v = u + a * t
4 m/s = u + 9.8 m/s² * t
Since we know that t = -u / a, we can substitute into the equation:
4 m/s = u + 9.8 m/s² * (-u / 9.8 m/s²)
4 m/s = u - u
4 m/s = 0
This indicates that the initial velocity (u) is 0 m/s. Plugging this value into the equation for height:
h = u² / (2 * 9.8 m/s²)
h = 0 / (2 * 9.8 m/s²)
h = 0 m
Now we can substitute the values of h and the gravitational potential energy into the equation and solve for the mass (m):
0.4 J = m * 9.8 m/s² * 0 m
0.4 J = 0 m²/s² * m
0.4 J = 0 m² * m / s²
0.4 J = 0 m³/s²
From the equation we can see that the mass value cannot be determined with the given information.
To find the mass of the ball, we can use the conservation of energy principle, which states that the total energy of a system remains constant.
Given that the ball's gravitational potential energy at its peak height is 0.4 joules, we know that it will gain an equal amount of kinetic energy as it falls back down. Hence, the ball will have a kinetic energy of 0.4 joules when it hits the ground.
The kinetic energy of an object can be calculated using the formula:
K.E. = 0.5 * mass * velocity^2
Therefore, we can rearrange the formula to solve for the mass:
mass = (2 * K.E.) / velocity^2
In this case, K.E. = 0.4 joules and velocity = 4 meters per second. Let's substitute these values into the formula to calculate the mass of the ball:
mass = (2 * 0.4) / (4^2)
mass = 0.8 / 16
mass = 0.05 kilograms
Hence, the mass of the ball is 0.05 kilograms.
(1/2) M V^2 = 0.4 J
You know V. Solve for M.