A cannonball of mass 3.67 kg is shot from a cannon at an angle of 39.51° relative to the horizontal and with an initial speed of 56.41 m/s. As the cannonball reaches the highest point of its trajectory, what is the gain in its potential energy relative to the point from which it was shot?
Well, let's see. When the cannonball reaches the highest point of its trajectory, its vertical velocity component becomes zero. So, it's at its highest point and we can calculate the gain in potential energy.
To calculate the gain in potential energy, we need to consider the change in height. At the highest point, the vertical displacement is the maximum. Now, I could go into all the complicated math, or I could just make a really bad joke about heights. What do you think?
To find the gain in potential energy of the cannonball at the highest point of its trajectory, we need to calculate the change in height.
The initial potential energy of the cannonball is zero since it is measured relative to the point it was shot from.
At the highest point of the trajectory, the vertical component of the velocity (v_y) is zero, and the object has reached its maximum height.
The vertical motion of the cannonball can be described using the kinematic equation:
v_y^2 = v_y0^2 + 2aΔy,
where v_y = 0 m/s (final vertical velocity), v_y0 = sin(theta) * v_0 (initial vertical velocity), a = -9.8 m/s^2 (acceleration due to gravity), and Δy is the change in height.
Since v_y = 0, we can solve for Δy:
0 = (sin(theta) * v_0)^2 + 2(-9.8)Δy.
Simplifying the equation:
0 = (sin(39.51°) * 56.41 m/s)^2 + (-19.6 m/s^2)Δy.
Now we can solve for Δy:
0 = (0.625 * 56.41)^2 + (-19.6)Δy.
0 = 22.79 + (-19.6)Δy.
19.6Δy = 22.79.
Δy = 22.79 / 19.6.
Δy ≈ 1.16 m.
Therefore, the gain in potential energy of the cannonball at the highest point of its trajectory is approximately 1.16 meters.
To find the gain in potential energy relative to the point from which it was shot, we need to calculate the potential energy at the highest point of the trajectory and subtract the initial potential energy.
1. Calculate the initial potential energy:
The initial potential energy of the cannonball relative to the point from which it was shot is zero because it is at the same level as the starting point.
2. Calculate the potential energy at the highest point of the trajectory:
At the highest point, the velocity of the cannonball becomes zero, and it reaches its maximum height. The change in potential energy is equal to the gain in potential energy.
To find the maximum height, we can use the kinematic equation for vertical motion:
v_f^2 = v_i^2 + 2as
Here, v_f is the final velocity (which is 0 m/s at the highest point), v_i is the initial velocity (56.41 m/s), a is the acceleration due to gravity (-9.8 m/s^2, taking downward as negative), and s is the vertical displacement (maximum height).
Rearranging the equation, we get:
s = (v_f^2 - v_i^2) / (2a)
Substituting the values, we have:
s = (0 - 56.41^2) / (2 * -9.8)
Calculate to find the maximum height.
3. Calculate the gain in potential energy:
The gain in potential energy at the highest point is given by the formula:
∆PE = m * g * h
where m is the mass of the cannonball (3.67 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height.
Substitute the values to calculate the gain in potential energy.
Note: In the formula, we use g = 9.8 m/s^2 because potential energy is defined with respect to the earth's gravitational field.
Therefore, follow these steps to find the gain in potential energy relative to the point from which it was shot:
1. Calculate the maximum height using the kinematic equation for vertical motion: s = (v_f^2 - v_i^2) / (2a).
2. Calculate the gain in potential energy using the formula ∆PE = m * g * h, where ∆PE is the gain in potential energy, m is the mass of the cannonball, g is the acceleration due to gravity, and h is the maximum height.
Plug in the values and solve to find the answer.