You exert a force of a known magnitude F on a grocery cart of total mass m. The force you exert on the cart points at an angle θ below the horizontal. If the cart starts at rest, determine an expression for the speed of the cart after it travels a distance d. Ignore friction.
Express your answer in terms of the variables F, m, d, and θ.
To solve this problem, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration. In this case, the force exerted on the cart is not directly in the horizontal direction, so we need to break it down into horizontal and vertical components.
First, let's resolve the force into its horizontal and vertical components. The horizontal component of the force is given by Fx = F * cos(θ), and the vertical component of the force is given by Fy = F * sin(θ).
Since there is no friction, the only force acting on the cart is the horizontal component Fx. According to Newton's second law, the net force is equal to the mass of the object multiplied by its acceleration (F = m * a).
Since the initial velocity of the cart is zero, and the acceleration is constant, we can use the equation of motion to find the final velocity. The equation for the displacement is given by d = (1/2) * a * t², where t is the time taken to travel the distance d.
To find the time, we can use the equation v = u + a * t, where u is the initial velocity (zero), v is the final velocity, and a is the acceleration.
Combining these equations, we get:
d = (1/2) * a * t²
v = 0 + a * t
Rearranging the second equation to get t in terms of v:
t = v / a
Plugging this into the first equation:
d = (1/2) * a * (v/a)²
d = (1/2) * v² / a
Now, we can substitute the value of acceleration with Fx/m:
d = (1/2) * v² / (Fx/m)
d = (m/2) * (v²/Fx)
Since Fx = F * cos(θ), we can substitute for Fx:
d = (m/2) * (v²/(F * cos(θ)))
Finally, rearranging the equation to solve for v:
v² = (2 * d * F * cos(θ)) / m
v = sqrt((2 * d * F * cos(θ)) / m)
Therefore, the expression for the speed of the cart after it travels a distance d is:
v = sqrt((2 * d * F * cos(θ)) / m)