a car is moving on a horizontal frictionless road and reches base of an upslope . what should be the speed of the car at the base of the upslope so that it can just reaCH its top without external force applied to it.(length of slope=15m, angle of slope=30degree, mass of car=250kg, gravity=10m/s2)
h = 15*sin30 = 7.5 m. = Ver. ht. of slope.
Y^2 = Yo^2 + 2g*d
Yo^2 = Y^2 - 2g*d
Yo^2 = 0 - (-20)*7.5 = 150
Yo = 12.25 m/s=Initial Ver. component of velocity.
Vo=Yo / sin30 = 12.25 / sin30=24.5 m/s
@ bottom of slope.
To find the speed of the car at the base of the upslope so that it can just reach its top without any external force, we can make use of the principle of conservation of energy.
Step 1: Calculate the height of the upslope.
The height of the upslope can be calculated using the formula:
height = length of slope * sin(angle of slope)
height = 15m * sin(30°) = 7.5m
Step 2: Calculate the potential energy at the base of the upslope.
The potential energy at the base of the upslope can be calculated using the formula:
potential energy = mass of car * gravity * height
potential energy = 250kg * 10m/s^2 * 7.5m = 18,750 Joules
Step 3: Calculate the velocity of the car at the base of the upslope.
The kinetic energy and potential energy at the base of the slope must be equal for the car to just reach the top without any external force applied. Therefore, we can equate the two energies and solve for velocity.
Kinetic energy = 1/2 * mass of car * velocity^2
18,750 Joules = 1/2 * 250kg * velocity^2
Divide both sides by 125kg:
150 Joules = velocity^2
Take the square root of both sides to find the velocity:
velocity = √150 Joules
velocity ≈ 12.25 m/s
Therefore, the speed of the car at the base of the upslope should be approximately 12.25 m/s in order to just reach the top without any external force applied.
To find the speed of the car at the base of the upslope so that it can just reach the top without any external force, we can make use of the principles of conservation of energy.
At the base of the upslope, we can consider the car to have two types of energy: kinetic energy due to its motion and potential energy due to its height above the ground.
The potential energy at the top of the slope should be equal to the kinetic energy at the base of the slope for the car to reach the top.
Let's break down the problem step by step:
Step 1: Find the potential energy at the top of the slope.
The potential energy (PE) is given by the formula: PE = m * g * h, where m is the mass, g is the acceleration due to gravity, and h is the height above the reference point (usually the ground). In this case, the reference point is the base of the upslope, and the height is the vertical distance from the base to the top, which is equal to h = length of slope * sin(angle of slope).
PE_top = m * g * h
Step 2: Find the potential energy at the base of the slope.
Since the car is at ground level at the base of the slope, the potential energy is zero.
PE_base = 0
Step 3: Find the kinetic energy at the base of the slope.
The kinetic energy (KE) is given by the formula: KE = (1/2) * m * v^2, where v is the velocity of the car.
KE_base = (1/2) * m * v_base^2
Step 4: Equate the potential energy at the top to the kinetic energy at the base.
PE_top = KE_base
m * g * h = (1/2) * m * v_base^2
Step 5: Solve for the velocity at the base of the upslope.
Rearrange the equation and solve for v_base:
v_base = sqrt(2 * g * h)
Now let's substitute the given values into the equation:
g = 10 m/s^2
h = length of slope * sin(angle of slope) = 15 * sin(30) = 7.5 m
v_base = sqrt(2 * 10 * 7.5) = sqrt(150) ≈ 12.25 m/s
Therefore, the speed of the car at the base of the upslope should be approximately 12.25 m/s for it to just reach the top without any external force.