A motorcyclist is trying to leap across the canyon by driving horizontally off the cliff at a speed of 38.0 m/s, height of the first cliff=50m, second cliff=30m.Ignoring the air resistance, find the speed with which the cycle strikes the ground on the other side.
starting K.E + gravitational P.E. = final K.E.
(1/2 * m * 38.0^2) + [m * 9.8 * (50 - 30)] = 1/2 * m * v^2
dividing by 1/2 m ... 38.0^2 + (2 * 9.8 * 20) = v^2
To find the speed with which the motorcycle strikes the ground on the other side of the canyon, we can use the principle of conservation of energy.
Here's how you can solve it step by step:
Step 1: Determine the potential energy at the top of the first cliff.
The potential energy (PE) is given by the formula:
PE = m * g * h,
where m is the mass of the motorcycle, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height of the first cliff (50 m).
Step 2: Calculate the kinetic energy at the top of the first cliff.
Since the motorcyclist is driving horizontally, there is no vertical motion. Hence, the kinetic energy (KE) at the top of the first cliff is equal to the potential energy:
KE1 = PE = m * g * h.
Step 3: Calculate the kinetic energy just before reaching the second cliff.
The kinetic energy at any point can be calculated using the formula:
KE = 0.5 * m * v²,
where m is the mass of the motorcycle and v is the velocity.
Since there is no change in potential energy, the kinetic energy just before reaching the second cliff (KE2) is equal to the kinetic energy at the top of the first cliff:
KE2 = KE1.
Step 4: Find the velocity just before reaching the second cliff.
Using the kinetic energy equation, we can write:
0.5 * m * v² = m * g * h.
Canceling the mass on both sides of the equation, we get:
0.5 * v² = g * h.
Simplifying further, we have:
v² = 2 * g * h.
Taking the square root of both sides, we find:
v = √(2 * g * h).
Step 5: Calculate the speed with which the cycle strikes the ground on the other side.
The speed is the magnitude of the velocity. Since the motorcyclist is driving horizontally, the velocity and speed will be the same. Therefore, the speed (v') is simply equal to the velocity just before reaching the second cliff:
v' = v = √(2 * g * h).
Now we can plug in the values:
g = 9.8 m/s² (acceleration due to gravity)
h = 30 m (height of the second cliff)
Calculating the speed, we have:
v' = √(2 * 9.8 * 30)
= √(588)
≈ 24.2 m/s.
So, the speed with which the motorcycle strikes the ground on the other side is approximately 24.2 m/s.