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.
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To determine the expression for the speed of the cart after it travels a distance d, we can use the work-energy principle. The work done on an object is equal to the change in its kinetic energy. Similarly, the work done on the grocery cart can be expressed as the force applied on it multiplied by the distance it travels, and the change in kinetic energy can be expressed as the final kinetic energy minus the initial kinetic energy.
The work done on the cart is given by the formula: Work = Force * Distance * cos(θ)
This is because the force component in the direction of motion is given by F * cos(θ).
The change in kinetic energy of the cart is given by the formula: ΔKE = (1/2) * m * v^2 - 0
Since the cart starts at rest, the initial kinetic energy is zero.
By equating the work done and the change in kinetic energy, we have:
Force * Distance * cos(θ) = (1/2) * m * v^2
Solving for v, the speed of the cart, we get:
v^2 = (2 * Force * Distance * cos(θ)) / m
Taking the square root of both sides, we have:
v = sqrt( (2 * Force * Distance * cos(θ)) / m )
Therefore, the expression for the speed of the cart after traveling a distance d is:
v = sqrt( (2 * F * d * cos(θ)) / m )