A child slides down a 5.98 m playground slide that makes an angle of 38.0° with the horizontal. The coefficient of kinetic friction between the slide and the child is 0.162. If she starts at the top with a speed of 0.208 m/s, what is her speed (in m/s) at the bottom?
net force of gravity of child down the slide: force=mgSin38-mg(.162)cos38
finalKEnergy=InitialKE+initial PE - work friction
1/2 m vf^2=1/2 m(.208^2)+mg*5.98*sin38-mg(.162)cos38
solve for vf.
This question is hard to solve without a diagram.
Draw a line of approximately 38° above the horizontal, so that it represents the slide. Next, draw a vertical arrow with its tail connected to the angled line about halfway down to represent the child’s weight (mg). Where the arrow and the angled line meet, draw a second arrow with its tail connected to the intersection and perpendicular to the angled line. This second arrow represents the normal force. A third arrow to represent the force of friction is pointing upwards along the slide.
To make it easier to solve, we are going to turn our usual x and y axes at an angle to match the slide. y is going to be in the direction of the normal force and x is going to point along the slide.
Resolve the the force of gravity (mg) into its components using these axes. You don't know "m" so just leave it as m. e.g. force of gravity is 9.8m It will cancel out later
Use the "y" component of gravity as the normal force to find friction.
Use the "x" component of gravity and the frictional force to find the net force. Now you can cancel out m and you are left with a value for acceleration. The rest is a kinematics problem.
Sorry, I was typing up an answer and didn't see that there was already a response.
To find the speed of the child at the bottom of the slide, we can first find the acceleration of the child along the slide using the force of gravity and the horizontal component of the normal force. Then we can use the kinematic equation to find the final speed.
1. Find the acceleration along the slide:
- The force of gravity acting on the child is given by the weight of the child, which can be calculated as:
Weight = mass × gravitational acceleration
In this case, we do not have the child's mass, but we can assume a standard value of 9.8 m/s² for gravitational acceleration.
- The horizontal component of the normal force counteracts the component of the weight force parallel to the slide and provides the frictional force.
Horizontal component of the normal force = normal force × cos(angle of the slide)
- The frictional force can be calculated as:
Frictional force = coefficient of kinetic friction × normal force
- The net force along the slide can be given as:
Net force along the slide = horizontal component of the normal force - Frictional force
- Finally, we can calculate the acceleration along the slide as:
Acceleration along the slide = Net force along the slide / mass
2. Use the kinematic equation to find the final speed:
- We can use the equation:
v_f² = v_i² + 2aΔx
where v_f is the final velocity, v_i is the initial velocity, a is the acceleration, and Δx is the distance traveled along the slide.
- In this case, the initial velocity is given as 0.208 m/s, the acceleration along the slide is the value calculated in step 1, and the distance traveled along the slide is given as 5.98 m. We need to solve for the final velocity, v_f.
Now let's go through the calculations step by step:
Step 1: Calculate the acceleration along the slide:
Weight = 9.8 m/s²
Horizontal component of the normal force = Weight × cos(38.0°)
Frictional force = 0.162 × Weight
Net force along the slide = Horizontal component of the normal force - Frictional force
Acceleration along the slide = Net force along the slide / mass (assuming mass = 1 since it cancels out in the calculation)
Step 2: Use the kinematic equation to find the final speed:
v_f² = v_i² + 2 × acceleration along the slide × distance traveled along the slide
Solve for v_f.
By plugging in the given values and performing the calculations, you can find the speed of the child at the bottom of the slide.