Question

A $75.2$ -kg man is riding on a $38.6-\mathrm{kg}$ cart traveling at a speed of $2.33\mathrm{m}/\mathrm{s}.$ He jumps off in such a way as to land on the ground with zero horizontal speed. Find the resulting change in the speed of the cart.

Short answer

The resulting change in speed of the cart can be found by calculating the initial and final total momentum, then subtracting the initial speed of the cart from the final speed.

Step-by-step solution

Determine the Initial Total Momentum

Calculate the initial momentum before the man jumps off the cart. This can be done by multiplying the total mass (man + cart) by the initial velocity. The equation goes as follows: $\text{{Initial momentum}}=(75.2kg+38.6kg)\times 2.33m/s$.

Determine the Final Total Momentum

Calculate the total momentum after the man jumps off the cart. Since the man lands on the ground with zero horizontal speed, his final momentum is zero. The final momentum is then only consisting of the momentum of the cart alone, which has a mass of $38.6$ kg and velocity $V$. The equation would be: $\text{{Final momentum}}=38.6kg\times V$.

Apply the Conservation of Momentum

Set the initial total momentum equal to the final total momentum, then solve for $V$. This will give you the final velocity of the cart.

Find the Change in Speed of the Cart

Subtract the initial speed from the final speed which you have calculated in the previous step. This will give you the change in speed of the cart.

Key concepts

Momentum

Momentum, a key concept in physics, represents the quantity of motion an object has. Mathematically, it is defined as the product of an object's mass and its velocity, given as the equation: $p=mv$, where $p$ is momentum, $m$ is mass, and $v$ is velocity. To intuitively understand momentum, you might imagine it as how much 'oomph' an object has when moving; a heavy truck at speed free-riding on a highway has a large momentum.

In our exercise, we calculate momentum to understand how the action of a man jumping off a cart affects the speed of the cart. The man and the cart initially share a certain momentum based on their combined mass and speed. Once the man jumps, he removes his share of that 'oomph,' altering the cart's speed due to conservation of momentum.

Inelastic Collisions

Inelastic collisions are events where colliding bodies come together and do not separate, contrasted with elastic collisions where objects bounce off one another retaining their kinetic energy. In the case of inelastic collisions, part of the kinetic energy is converted into other forms of energy, like heat or sound, hence the total kinetic energy is not conserved.

However, an essential concept relevant to our exercise is that, even in inelastic collisions, momentum is conserved. In the exercise, although no collision is described, the man's action is akin to an inelastic event, where he moves from the cart to the ground with zero horizontal speed, similar to how objects stick together after a collision. The total system momentum before the event is equal to the total momentum after the event, which directs us to calculate the final speed of the cart.

Physics Problem Solving

Solving physics problems requires a systematic approach: understanding the problem, identifying known and unknown variables, selecting relevant principles (like conservation of momentum), and applying appropriate equations to find a solution.

In our example, we first identified the total initial momentum of the man-cart system. After the man jumps off, knowing his final momentum was zero (since he lands with no horizontal speed), we only had to consider the cart's momentum. By applying the conservation of momentum principle (initial momentum equals final momentum), we found the final velocity of the cart and then the change in its speed. This problem-solving approach is a powerful tool across various physics problems, guiding us from initial conditions through the conservation laws to the final solution.