A skateboarder at a skate park rides along the path shown in (Figure 1) .a=2.7 m and b=1.0m
Part A
If the speed of the skateboarder at point A is v=1.6m/s, what is her speed at point B? Assume that friction is negligible.
To determine the skateboarder's speed at point B, we can utilize the principle of conservation of energy.
Here's how we can approach it:
Step 1: Identify the points A and B on the path.
Step 2: Recognize that the skatepark has no significant friction, meaning energy is conserved. This means that the sum of the kinetic energy and the potential energy at any point on the path must remain constant.
Step 3: Calculate the kinetic energy at point A using the formula:
KE_A = (1/2) * m * v_A^2
Where:
- KE_A is the kinetic energy at point A,
- m is the mass of the skateboarder (which we assume to be constant),
- v_A is the speed at point A.
Step 4: Use the conservation of energy principle to find the kinetic energy at point B. Since there is no potential energy at point B (since it is at the same height as A), the kinetic energy at point B is equal to the kinetic energy at point A:
KE_B = KE_A
Step 5: Substitute the expression for kinetic energy at point A (from step 3) into the equation from step 4:
(1/2) * m * v_A^2 = (1/2) * m * v_B^2
Step 6: Solve for v_B, the speed at point B:
v_B^2 = (v_A^2 * b^2) / a^2
Step 7: Substitute the given values for v_A, a, and b into the equation from step 6 and solve for v_B:
v_B^2 = (1.6^2 * 1.0^2) / 2.7^2
v_B^2 = 0.951
Taking the square root of both sides, we find:
v_B ≈ 0.975 m/s
Therefore, the skateboarder's speed at point B is approximately 0.975 m/s.
To determine the speed of the skateboarder at point B, we can use the principle of conservation of mechanical energy. According to this principle, the total mechanical energy of the skateboarder remains constant along the path, assuming no external forces act on her.
The total mechanical energy is given by the sum of the kinetic energy and the potential energy at each point. Mathematically, it can be expressed as:
E = KE + PE
Where:
E is the total mechanical energy
KE is the kinetic energy
PE is the potential energy
Since we are assuming no friction, the only form of energy the skateboarder has is kinetic energy. Therefore, at point A, the total mechanical energy is equal to the kinetic energy:
E_A = KE_A
At point B, the total mechanical energy is once again equal to the kinetic energy:
E_B = KE_B
Since E_A = E_B, we can equate the kinetic energies:
KE_A = KE_B
The formula for kinetic energy is:
KE = 0.5 * m * v^2
Where:
KE is the kinetic energy
m is the mass of the skateboarder (which we assume to be constant)
v is the velocity/speed of the skateboarder
At point A, the velocity v_A is given as 1.6 m/s. We can use this information to calculate the kinetic energy at point A. Then, we can rearrange the formula and solve for v_B, the velocity at point B.
KE_A = 0.5 * m * v_A^2
Simplifying:
KE_A = 0.5 * m * (1.6)^2
Now, we can set this equal to the kinetic energy at point B:
KE_A = KE_B
0.5 * m * (1.6)^2 = 0.5 * m * v_B^2
Cancelling out the mass m:
(1.6)^2 = v_B^2
Simplifying:
2.56 = v_B^2
Taking the square root of both sides:
√2.56 = √(v_B^2)
1.6 = v_B
Therefore, the speed of the skateboarder at point B is 1.6 m/s.