60-kg skier passes point a with a speed of 5 m/s, determine his speed when he reaches point b. also find the normal force exerted on him by the slope at this point.
To determine the skier's speed at point b and the normal force exerted on him by the slope at that point, we need to consider conservation of energy.
1. Start by determining the potential energy (PE) and kinetic energy (KE) of the skier at point a.
The potential energy at point a is given by:
PE_a = m * g * h_a
where m is the skier's mass (60 kg), g is the acceleration due to gravity (9.8 m/s^2), and h_a is the height of point a above a reference point.
Since we don't have the height information, we assume point a is at the same height as point b (h_a = h_b). Therefore, the potential energy at point a can be written as:
PE_a = m * g * h_b
The kinetic energy at point a is given by:
KE_a = (1/2) * m * v_a^2
where v_a is the velocity of the skier at point a (5 m/s).
2. Next, determine the potential energy and kinetic energy at point b.
The potential energy at point b is given by:
PE_b = m * g * h_b
The kinetic energy at point b is given by:
KE_b = (1/2) * m * v_b^2
where v_b is the velocity of the skier at point b (what we need to find).
3. Apply the conservation of energy.
According to the conservation of energy principle, the total mechanical energy of the skier remains constant throughout the motion. Therefore, the sum of the kinetic energy and potential energy at point a should be equal to the sum of the kinetic energy and potential energy at point b.
PE_a + KE_a = PE_b + KE_b
Substituting the expressions for potential energy and kinetic energy, we have:
(m * g * h_b) + (1/2) * m * v_a^2 = (m * g * h_b) + (1/2) * m * v_b^2
Simplifying the equation, we get:
(1/2) * m * v_a^2 = (1/2) * m * v_b^2
4. Solve for v_b.
Divide both sides of the equation by (1/2) * m:
v_a^2 = v_b^2
Taking the square root of both sides:
v_a = v_b
Therefore, the speed of the skier when he reaches point b is the same as the speed at point a, which is 5 m/s.
5. Determine the normal force exerted on the skier by the slope at point b.
At point b, the normal force (N) must balance the gravitational force acting on the skier. The normal force is perpendicular to the slope of the slope. If the slope has an angle of inclination (θ) with respect to the horizontal, then the normal force can be calculated as:
N = m * g * cos(θ)
In this case, since we don't have the information about the angle of inclination, we cannot directly determine the normal force.
To summarize:
The skier's speed when he reaches point b is 5 m/s.
The normal force exerted on the skier by the slope at point b cannot be determined without knowing the angle of inclination of the slope.