A skier reaches a speed of 42 m/s on a 25◦ski slope. Ignoring friction, what was the distance along the slope the skier would have had to travel, starting from rest?
To calculate the distance along the slope the skier would have had to travel, we need to use the equation of motion for an object on an inclined plane:
distance = (initial velocity^2 - final velocity^2) / (2 * acceleration)
In this case, the initial velocity is 0 m/s (starting from rest), the final velocity is 42 m/s, and the acceleration can be calculated using the formula:
acceleration = gravitational force * sin(theta)
where theta is the angle of the slope.
The gravitational force can be calculated using the formula:
gravitational force = mass * gravitational acceleration
Let's assume the mass of the skier to be 70 kg and the gravitational acceleration to be 9.8 m/s^2.
First, let's calculate the gravitational force:
gravitational force = 70 kg * 9.8 m/s^2 = 686 N
Next, let's calculate the acceleration:
acceleration = 686 N * sin(25°)
Now, we can substitute the values into the equation to find the distance:
distance = (0^2 - 42^2) / (2 * acceleration)
Once you calculate the value of acceleration, plug it into the equation and solve for distance.
To find the distance along the slope that the skier would have had to travel, we can use the basic principles of physics. Here's the step-by-step explanation of how to calculate it:
Step 1: Convert the angle from degrees to radians.
The given angle is 25°. To convert it to radians, we use the formula:
Radians = (degrees * π) / 180°
Radians = (25 * π) / 180°
Radians = 0.4363 radians (approx.)
Step 2: Calculate the vertical component of the skier's velocity.
Since the skier reaches a speed of 42 m/s, we can find the vertical component of the velocity using trigonometry.
Vertical Component = Velocity * sin(θ)
Vertical Component = 42 m/s * sin(0.4363)
Vertical Component = 42 m/s * 0.4259
Vertical Component = 17.8698 m/s (approx.)
Step 3: Calculate the time taken to reach this speed.
Since the skier starts from rest (initial velocity = 0), we can use the equation:
Final Velocity = Initial Velocity + (Acceleration * Time)
17.8698 m/s = 0 m/s + (9.8 m/s^2 * Time)
Time = 17.8698 m/s / 9.8 m/s^2
Time = 1.826 seconds (approx.)
Step 4: Calculate the horizontal distance traveled.
Now, we can find the horizontal distance traveled using the equation for distance:
Distance = Velocity * Time
Distance = 42 m/s * 1.826 seconds
Distance = 76.692 meters (approx.)
Therefore, the skier would have had to travel approximately 76.692 meters along the slope, starting from rest.
component of weight force down slope = m g sin 25
so
acceleration a = 9.81 sin 25 = 4.15 m/s^2
v = 4.15 t
42 = 4.15 t
t = 10.1 seconds
d = (1/2) a t^2
d = (1/2) *4.15 *10.1^2 = 840 meters