A skier (m=61.0 kg) starts sliding down from the top of a ski jump with negligible friction and takes off horizontally. If h = 7.70 m and D = 11.4 m, find H.
Find her total kinetic energy as she reaches the ground.
To find H, we can use the principle of conservation of mechanical energy. At the top of the ski jump, the skier has potential energy due to gravity, which is then converted into kinetic energy as the skier slides down.
1. Find the potential energy at the top of the ski jump:
Potential energy (PE) = mass (m) × gravitational acceleration (g) × height (h)
PE = 61.0 kg × 9.8 m/s^2 × 7.70 m
2. Find the initial kinetic energy (KE_i):
At the top of the ski jump, all the potential energy will be converted into kinetic energy, so KE_i = PE.
KE_i = 61.0 kg × 9.8 m/s^2 × 7.70 m
3. Find the final velocity (v_f):
Since the skier takes off horizontally, there is no vertical velocity at the takeoff point. Therefore, the final velocity in the vertical direction is 0 m/s.
4. Use the horizontal distance (D) to find the time of flight (t):
The horizontal distance (D) is given by: D = velocity (v_x) × time (t)
Since the horizontal velocity (v_x) is constant, we can rewrite the equation as: D = v_x × t
v_x = D / t
5. Use the vertical distance (h) to find the time of flight (t):
The vertical distance (h) is given by: h = (1/2) × gravitational acceleration (g) × t^2
Rearranging the equation to solve for t: t = sqrt(2h / g)
6. Substitute the value of t into the equation for v_x to find the horizontal velocity (v_x).
v_x = D / (sqrt(2h / g))
7. Find the final kinetic energy (KE_f):
The final kinetic energy (KE_f) can be calculated using the formula:
KE_f = (1/2) × mass (m) × (velocity (v_x))^2
To summarize:
1. Calculate PE = m × g × h.
2. Calculate KE_i = PE.
3. Calculate t = sqrt(2h / g).
4. Calculate v_x = D / t.
5. Calculate KE_f = (1/2) × m × (v_x)^2.
Now, to find the skier's total kinetic energy as she reaches the ground, substitute the known values into the final kinetic energy equation (KE_f).